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Anal Math Phys
Anal Math Phys
Analysis and Mathematical Physics
1664-2368
1664-235X
Springer International Publishing Cham

924
10.1007/s13324-024-00924-z
Article
Conjugations of unitary operators, I
Mashreghi Javad 1
Ptak Marek 2
Ross William T. wross@richmond.edu

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1 https://ror.org/04sjchr03 grid.23856.3a 0000 0004 1936 8390 Département de mathématiques et de statistique, Université Laval, Québec, QC G1K 0A6 Canada
2 https://ror.org/012dxyr07 grid.410701.3 0000 0001 2150 7124 Department of Applied Mathematics, University of Agriculture, ul. Balicka 253c, 30-198 Kraków, Poland
3 https://ror.org/03y71xh61 grid.267065.0 0000 0000 9609 8938 Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173 USA
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16 5 2024
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If U is a unitary operator on a separable complex Hilbert space H, an application of the spectral theorem says there is a conjugation C on H (an antilinear, involutive, isometry on H) for which CUC=U∗. In this paper, we fix a unitary operator U and describe all of the conjugations C which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for U if and only if it is invariant for any conjugation C for which CUC=U∗.

Keywords

Complex symmetric operators
Unitary operators
Mathematics Subject Classification

47B35
47B02
47A05
Ministry of Science and Higher Education of the Republic of Polandhttp://dx.doi.org/10.13039/501100002790 Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada issue-copyright-statement© Springer Nature Switzerland AG 2024
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pmcIntroduction

A version of the spectral theorem says that any unitary operator U on a separable complex Hilbert space H is unitarily equivalent to a multiplication operator Mφf=φf on a Lebesgue space L2(X,μ) [33, p. 13]. Here X is a compact Hausdorff space, μ is a finite positive Borel measure on X, and φ∈L∞(X,μ) is unimodular μ-almost everywhere. If J is the mapping from L2(X,μ) to itself defined by (Jf)(x)=f(x)¯, x∈X (the bar denotes complex conjugation), then J is an antilinear, isometric, and involutive map and is called a conjugation. Furthermore, the conjugation J induces the adjoint identity JMφJ=Mφ¯=Mφ∗. Via unitary equivalence, given any unitary operator U on H, this results in a conjugation C on H for which CUC=U∗ (see Lemma 2.3). In the parlance of operator theory, one says that U is a C-symmetric operator [18, 19]. As it turns out, there are many conjugations C on H for which CUC=U∗. The goal of this paper is to describe them all.

Towards this goal, letCs(U):={Cis a conjugation onH:CUC=U∗}.

As discussed in the previous paragraph, Cs(U)≠∅ (see also Proposition 2.5). On the other extreme, Cs(I), where I is the identity operator on H, is simply the set of all conjugations on H. The subscript s, for “symmetric”, in the above definition of Cs(U) might seem superfluous. However, in a follow up paper to this one [28], we will explore the set Cc(U) (notice the subscript c), the conjugations on H that commute with U, i.e., CUC=U. This paper provides several characterizations of Cs(U). One such characterization (Theorem 8.2) involves the spectral theorem. When one is fortunate enough to have a concrete spectral decomposition of a unitary operator, for example when U is an n×n unitary matrix, one can give a very tangible description of Cs(U). In particular, Theorem 4.2 says that if U is an n×n unitary matrix with spectral decompositionU=Wξ1In1ξ2In2⋱ξdIndW∗,

where W is an n×n unitary matrix, ξ1,…,ξd (1⩽d⩽n) are the distinct eigenvalues of U, Inj is the nj×nj identity matrix, and n1+n2+⋯+nd=n, then any conjugation C on Cn for which CUC=U∗ must take the form1.1 C=WV1V2⋱VdJW∗,

where each Vj is an nj×nj unitary matrix satisfying Vjt=Vj (t denotes the transpose of a matrix), and J denotes the conjugation on Cn defined byJ[x1x2⋯xn]t=[x1¯x2¯⋯xn¯]t.

A version of (1.1) can be obtained in the infinite dimensional setting when the eigenvectors for U are complete in H. In particular, we give a concrete description of Cs(U) when U is the Fourier–Plancherel transform on L2(R) (Example 4.4) and when U is the Hilbert transform on L2(R) (Example 4.6).

The main driver of nearly all the results in this paper is a measure-theoretic decomposition of any C∈Cs(U) given in Theorem 3.2. This enables us to examine manageable pieces of C of H on certain reducing subspaces of U.

Another description of Cs(U) relies less on the spectral decomposition of a unitary operator U, which is often quite intangible in the general setting, and more on some special properties of U. For example, if U is a bilateral shift (see Definition 6.1), Cs(U) was described in [5] (see also Theorem 6.4). In particular, when U is the bilateral shift (Mξf)(ξ)=ξf(ξ) on L2(m,T) (m is normalized Lebesgue measure on the unit circle T), then C∈Cs(U) if and only if C=MuJ, where J is the conjugation on L2(m,T) given by Jf=f¯ and u∈L∞(m,T) is unimodular almost everywhere on T (see Example 6.6).

The unitary multiplication operator Mψ on L2(m,T), where ψ is an inner function (a bounded analytic function on D whose radial boundary values on T have modulus one m-almost everywhere [10, 23]), turns out to be a bilateral shift [29, Proposition 5.17] and a concrete description of Cs(Mξ) is given in Theorem 7.8. In fact (see Remark 7.1), such Mψ serve as models for all bilateral shifts and so, in a way, Theorem 7.8, is a canonical model for Cs(U) for any bilateral shift U.

As a byproduct of several results in this paper, we can connect conjugations with the hyperinvariant subspaces of a unitary operator U on H (those subspace which are invariant for any bounded operator on H commuting with U). Certainly the invariant and hyperinvariant subspaces of unitary operators have been discussed before (for example [8, 9, 31]). Here we connect them with conjugations in the following way. For a (closed) subspace M of H, the following are equivalent: (i) M is hyperinvariant for U; (ii) CM⊆M for every C∈Cs(U). This result is obtained, along with several other equivalent conditions covered in this paper, via Theorems 3.2, 3.8, and 8.6 and is officially stated in Theorem 8.8.

Basic facts about conjugations and spectral measures

All Hilbert spaces H in this paper will be complex and separable. Let B(H) denote the set of all bounded linear transformations on H and AB(H) denote the set of all bounded antilinear transformations on H. By this we mean that C∈AB(H) when C(x+αy)=Cx+α¯Cy for all x,y∈H and α∈C (C is antilinear) and sup{‖Cx‖:‖x‖=1} is finite (C is bounded). We say that C∈AB(H) is a conjugation if C satisfies the two additional conditions ‖Cx‖=‖x‖ for all x∈H (C is isometric) and C2=I (C is involutive). By the polarization identity, a conjugation also satisfies2.1 ⟨Cx,Cy⟩=⟨y,x⟩forallx,y∈H.

Conjugations play an important role in operator theory and were initially studied in [13, 14, 17–19]. More recently, conjugations were explored in various settings and applications in [4, 5, 11, 12, 27, 32].

Example 2.2

Many types of conjugations were outlined in [17–19]. Below are a few basic ones that are relevant to this paper. As discussed in the introduction, the mapping Cf=f¯ defines a conjugation on L2(X,μ). In particular, the mapping C[x1x2⋯xn]t=[x1¯x2¯⋯xn¯]t defines a conjugation on Cn. Throughout this paper we will use the notation t to represent the transpose of a matrix. In addition, vectors in Cn will be viewed as column vectors since, for an n×n matrix A of complex numbers, we will often consider the linear transformations on Cn defined by x↦Ax.

One can consider the conjugations (Cf)(t)=f(t)¯ and (Cf)(t)=f(-t)¯ on L2(R). These were used in [1, 2] to study symmetric operators and their connections to physics.

If u is an inner function, H2 is the standard Hardy space, and Ku:=H2⊖uH2 is the model space associated with u (considering everything as a subspace of L2(m,T) via radial boundary values), then (Cf)(ξ)=u(ξ)ξf(ξ)¯ defines a conjugation on Ku [15, Ch. 8].

This next lemma enables us to transfer a conjugation on one Hilbert space to a conjugation on another. The (easy) proof is left to the reader.

Lemma 2.3

Suppose H and K are Hilbert spaces and V:H→K is a unitary operator. If C is a conjugation on H then VCV∗ is a conjugation on K.

For a conjugation C on H and an A∈B(H), we say that A is C-symmetric if CAC=A∗. A multitude of operators from a variety of settings enjoy this property [17, 18, 21, 22]. Here are a few examples that are relevant to this paper.

Example 2.4

If φ∈L∞(X,μ) and Mφ denotes the multiplication operator Mφf=φf on L2(X,μ), then the conjugation C from Example 2.2(a) satisfies CMφC=Mφ¯=Mφ∗. Thus, Mφ is C-symmetric. For a normal operator N∈B(H), the spectral theorem says that N is unitarily equivalent to Mφ on L2(X,μ) for some finite positive Borel measure on some compact Hausdorff space X. Now use the discussion above and Lemma 2.3 to see that N is a C-symmetric operator for some conjugation C on H.

The conjugation (Cf)(x)=f(-x)¯ from Example 2.2(b) makes the translation operator (Uf)(x)=f(x-1) on L2(R) a C-symmetric unitary operator. This conjugation also makes the Hilbert transform a C-symmetric unitary operator.

The conjugation (Cf)(t)=f(t)¯ from Example 2.2(b) makes the Fourier–Plancherel transform a C-symmetric unitary operator on L2(R).

Though not used in this paper, but certainly in our follow up paper [28], we recall the following result from [24] which shows that any unitary operator can be built from conjugations. We include a short proof (slightly different from the original) for the reader’s convenience.

Proposition 2.5

For each unitary operator U on H, there are conjugations J1 and J2 on H such that U=J1J2. Moreover, J1,J2∈Cs(U).

Proof

For a given unitary U on H, we can use the spectral theorem argument from Example 2.4(a) (and also from the introduction) to produce a conjugation J1 on H such that J1UJ1=U∗. Now observe that J2:=U∗J1 is a conjugation, J2UJ2=U∗, and J1J2=U.

The following result will be used from time to time in this paper.

Proposition 2.6

Suppose U, V, W are unitary operators on H such that WUW∗=V. Then WCs(U)W∗=Cs(V).

The following result from [20, Lemma 3.2], which will be generalized below, gives an explicit description of all the conjugations on Cn.

Proposition 2.7

A mapping C on Cn is a conjugation if and only if C=VJ, where V is an n×n unitary matrix with Vt=V and J is the conjugation on Cn defined by J[x1x2⋯xn]t=[x1¯x2¯⋯xn¯]t, i.e.,C[x1x2⋯xn]t=V[x1¯x2¯⋯xn¯]t.

Our discussion below needs a generalization of the previous result for Cn to an abstract (separable) Hilbert space H, but, of course, we need a substitute for Vt since the “transpose” of a linear operator V on H needs a proper definition (as opposed to the adjoint V∗ which has a clear and well understood definition). For a fixed orthonormal basis B={uj}j⩾1 for H and an A∈B(H), we use the notation[A]B=[amn]m,n⩾1=[⟨Aun,um⟩]m,n⩾1

to denote the matrix representation of A with respect to B, considered as a bounded operator on the sequence space2.8 ℓ+2:=x=[x1x2⋯]t,xj∈C:‖x‖ℓ+2=∑j⩾1|xj|21/2<∞

by x↦[A]Bx. For this fixed orthonormal basis B, we also define a conjugation JB on H by2.9 JB∑j⩾1cjuj=∑j⩾1cj¯uj.

In other words, JB fixes every basis element uj and extends antilinearly to H. Here is a version of Proposition 2.7 for a general, possibly infinite dimensional, separable Hilbert space.

Proposition 2.10

For an orthonormal basis B={uj}j⩾1 for a separable Hilbert space H, the mapping C is a conjugation on H if and only if there is a unitary operator V on H such that [V]Bt=[V]B and C=VJB.

Proof

Suppose C is a conjugation on H. Then V=CJB is a unitary operator on H (since it is linear, isometric, and onto). Moreover, applying (2.9) and then (2.1), we see that⟨Vum,un⟩=⟨CJBum,un⟩=⟨Cum,un⟩=⟨Cun,um⟩=⟨CJBun,um⟩=⟨Vun,um⟩.

Thus [V]Bt=[V]B and C=VJB.

Conversely, if V is a unitary operator on H with [V]Bt=[V]B, define C=VJB. Observe that C is antilinear and isometric. We just need to verify that C2=I. Since C is antilinear, note that C2=VJBVJB is a bounded linear operator on H. Moreover, since JBVJB is also a bounded linear operator on H, we see that2.11 [C2]B=[V]B[JBVJB]B.

Now observe that for all m,n⩾1, again making use of (2.1) and (2.9),⟨JBVJBum,un⟩=⟨JBVum,un⟩=⟨JBun,Vum⟩=⟨un,Vum⟩=⟨Vum,un⟩¯.

Thus, from our assumption that [V]Bt=[V]B, we see that2.12 [JBVJB]B=[V]B¯=[V]Bt¯=[V∗]B.

From (2.11) it follows that [C2]B=[I]B and so C2=I. Thus, C is a conjugation on H.

Remark 2.13

If J is any conjugation on H, one can show there exists an orthonormal basis B for H such that J=JB [18, Lemma 1]. Thus, there is some freedom in the above analysis, if so desired, to choose a conjugation J first instead of the orthonormal basis B.

A version of the spectral theorem for unitary operators (see [7, Ch. IX, Thm. 2.2] or [33, Ch. 1]) says that if U is a unitary operator on H, then there is a unique spectral measure E(·) defined on the Borel subsets of σ(U), the spectrum of U, such that2.14 U=∫σ(U)ξdE(ξ).

Moreover, for any spectral measure E(·) on T (i.e., E(·) is a projection-valued function on the Borel subsets of T such that E(T)=I and E(·) is countably additive), there is a unique unitary operator U associated with E(·) via (2.14).

For a spectral measure E(·) and x,y∈H, the function μx,y(·):=⟨E(·)x,y⟩ defines a finite complex Borel measure on T and, in particular, for each x∈H,2.15 μx:=μx,x

defines a positive finite Borel measure on T, sometimes called an elementary measure.

The following proposition from [24] describes the relationship between a unitary operator, its spectral measure, and a conjugation.

Proposition 2.16

For a conjugation C and a unitary operator U on H with associated spectral measure E(·), the following are equivalent. C∈Cs(U);

CE(Ω)C=E(Ω) for every Borel set Ω⊆σ(U).

Decompositions of conjugations

As we will see in subsequent sections, the set Cs(U) is quite large and so an important step in understanding it is to decompose each C∈Cs(U) into more manageable pieces. This decomposition will involve various types of invariant subspaces. Recall that a (closed) subspace M of a Hilbert space H is invariant for A∈B(H) if AM⊆M; reducing if AM⊆M and A∗M⊆M; and hyperinvariant if TM⊆M for every T∈B(H) that commutes with A. We begin with the following lemma which follows from (2.1) and the fact that C2=I.

Lemma 3.1

If C is a conjugation on H and M is a C-invariant subspace of H, then CM=M and CM⊥=M⊥.

Below is a useful decomposition theorem for a conjugation. For the rest of this paper, M+(T) will denote the set of all finite positive Borel measures on the unit circle T. For μ,σ∈M+(T) we use the standard notation μ≪σ for μ is absolutely continuous with respect to σ and μ⊥σ for μ is singular with respect to σ.

Theorem 3.2

Let U be a unitary operator on H, E(·) be its associated spectral measure, and for each x∈H, let μx denote the associated elementary measure from (2.15). For any μ∈M+(T), letHμ:={x∈H:μx≪μ}.

Then we have the following. Hμ is a reducing subspace for U.

If C∈Cs(U), then (i) CHμ=Hμ and CHμ⊥=Hμ⊥.

(ii) C=Cμ⊕Cμ⊥ where Cμ:=C|Hμ and Cμ⊥:=C|Hμ⊥.

(iii) Cμ(U|Hμ)Cμ=U∗|Hμ and Cμ⊥(U|Hμ⊥)Cμ⊥=U∗|Hμ⊥.

Proof

The proof of (a) is routine and we omit it (see [25, §65, §66] for the details).

To prove (b), let x∈Hμ and Ω be a Borel subset of σ(U) for which μ(Ω)=0. By Proposition 2.16, C commutes with E(Ω) and thus, via (2.1),μCx=⟨E(Ω)Cx,Cx⟩=⟨CE(Ω)x,Cx⟩=⟨x,E(Ω)x⟩=μx(Ω)=0.

This proves that Cx∈Hμ. Since C is a conjugation, we see that CHμ=Hμ and CHμ⊥=Hμ⊥ (Lemma 3.1). This proves (i) and (ii). Part (iii) follows from the facts that Hμ is a reducing subspace for U and C∈Cs(U).

As a note here, other properties of the reducing subspace Hμ were explored by Halmos in [25, §65]. Observe that if α∈T and δα denotes the point mass at α, one can see thatHδα=ker(U-αI).

Corollary 3.3

For a unitary operator U on H, let C∈Cs(U). Suppose α∈T. Then C=Cδα⊕Cδα⊥, where Cδα∈Cs(U|ker(U-αI)) and Cδα⊥∈Cs(U|ker(U-αI)⊥).

The following corollary will play an important role later on.

Corollary 3.4

If U is a unitary operator on H withH=⨁j⩾1Eξj,Eξj:=ker(U-ξjI),

thenC=⨁j⩾1C|Eξj

andC|EξjU|EξjC|Eξj=U∗|Eξjforallj⩾1.

Remark 3.5

Apply Theorem 3.2 to the case when μ=m (normalized Lebesgue measure on T) and let Hac:=Hm and Hsing:=Hm⊥. The decomposition H=Hac⊕Hsing was first explored in [25] and considered further in [3, 29, 30]. One can prove the following: Let U be a unitary operator on H and let C∈Cs(U). If H=Hac⊕Hsing is the Lebesgue decomposition from above, then C=Cac⊕Csing, where Cac∈Cs(U|Hac) and Csing∈Cs(U|Hsing).

A von Neumann–Wold type decomposition surveyed in [29] yields the following decomposition of a unitary operator U on H by reducing subspaces Hac, L, and ⨁α∈Tker(U-αI) such that3.6 H=Hac⨁L⨁α∈Tker(U-αI).

Here one proves the following: Let U be a unitary operator on H and C∈Cs(U). If H is decomposed as in (3.6), then there is an appropriate decomposition of C as C=Cac⊕C|L⊕⨁α∈TCδα, where each summand is a conjugation on the corresponding space.

Though Theorem 3.2 seems relatively simple, it yields interesting results about hyperinvariant subspaces of unitary operators. For a unitary operator, every hyperinvariant subspace is also reducing (but not the converse as one can see with a diagonal matrix). Theorem 3.8 below connects hyperinvariant subspaces with conjugations. The culmination of this discussion will be given in Theorems 8.6 and 8.8.

Theorem 3.7

Let U be a unitary operator on H with spectral measure E(·). Then every hyperinvariant subspace for U is invariant for every C∈Cs(U). Moreover, every hyperinvariant subspaces can be written as E(Ω)H, where Ω is a Borel subset of σ(U).

Proof

From [33, Proposition 6.9] (see also [8]), every hyperinvariant subspace for U can be written as E(Ω)H for some Borel set Ω⊆σ(U). Since C commutes with E(·) (Proposition 2.16(b)), E(Ω)H is invariant for C.

We now connect Hμ with hyperinvariant subspaces. A first step in this direction is this following result – which might be contained somewhere in the literature but we include a proof for the reader’s convenience. We will see an extension of this result in Theorems 8.6 and 8.8 below.

Theorem 3.8

If M is a hyperinvariant subspace of a unitary operator U on H, then there is a μ∈M+(T) for which M=Hμ.

Proof

From Theorem 3.7, every hyperinvarient subspace M of U, with associated spectral measure E(·), takes the form M=E(Ω)H for some Borel subset Ω⊆σ(U). Since E(Ω)H is also reducing, then V:=U|E(Ω)H is a unitary operator whose spectral measure is E(·)E(Ω).

Let W∗(V) denote the von Neumann algebra generated by V (i.e., the smallest ∗-closed and strongly closed algebra containing V). Using [7, Ch. IX, Cor. 7.9] one can produce a separating vector e∈H for W∗(V) in that if A∈W∗(V) satisfies Ae=0, then A≡0. By [7, Ch. IX, Prop. 8.3], the corresponding elementary measure μe from (2.15) is a scalar-valued spectral measure for V in that μe(Δ)=0 if and only if E(Δ)E(Ω)=E(Δ∩Ω)=0.

To complete the proof, we will show that E(Ω)H=Hμe. For the ⊆ containment, observe that if x∈E(Ω)H and Δ is a Borel subset of Ω, then3.9 μx(Δ)=⟨E(Δ)x,x⟩=‖E(Δ)x‖2

and thus, since μe is a scalar valued spectral measure for V=U|E(Ω)H, we see that if μe(Δ)=0, then E(Δ)=0. Thus, by (3.9), μx≪μe and so x∈Hμe. This verifies E(Ω)H⊆Hμe.

For the ⊇ containment, let y∈Hμe. Then μy≪μe and so μy(Δ)=0 whenever μe(Δ)=0. We want to show that y∈E(Ω)H. Since μe is a scalar valued spectral measure for V, we see that μe(σ(U)\Ω)=0. Using the assumption that μy≪μe, we obtainμy(σ(U)\Ω)=‖E(σ(U)\Ω)y‖2=0.

Thus, since H=(E(Ω)H)⊕(E(σ(U)\Ω)H), it follows that y∈E(Ω)H. This verifies Hμe⊆E(Ω)H.

We end this section with a discussion of when Hν1⊆Hν2 for ν1,ν2∈M+(T). Observe that if ν is a scalar spectral measure for U (i.e., ν(Δ)=0 if and only if E(Δ)=0) then Hν=H. Next let[x]U,U∗:=⋁{Unx:n∈Z},

where ⋁ denotes the closed linear span, denote the ∗-cyclic invariant subspace generated by x. For any μ∈M+(T) the condition x∈Hμ implies that [x]U,U∗⊆Hμ (since Hμ is reducing).

Recall [25, §48] the standard Boolean operations ∧ and ∨ for μ1,μ2∈M+(T) defined on Borel subsets Ω of T by(μ1∨μ2)(Ω)=μ1(Ω)+μ2(Ω);(μ1∧μ2)(Ω)=inf{μ1(Ω∩A)+μ2(Ω\A):Ais a Borel set}.

Proposition 3.10

Let U be a unitary operator on H and μ be any scalar spectral measure for U. For ν1,ν2∈M+(T) the following are equivalent. Hν1⊆Hν2;

ν1∧μ≪ν2∧μ.

Proof

Assume condition (b) and let x∈Hν1. Then ⟨E(·)x,x⟩≪ν1. Since μ is a scalar spectral measure for U, we see that ⟨E(·)x,x⟩≪μ. Hence ⟨E(·)x,x⟩≪ν1∧μ≪ν2∧μ. Thus ⟨E(·)x,x⟩≪ν2 and so x∈Hν2. Thus, (b)⟹(a).

For the proof of (a)⟹(b), let us assume that ν1∧μ is not absolutely continuous with respect to ν2∧μ. Then there is a nonzero measure ν∈M+(T) such that ν≪(ν1∧μ) and ν⊥(ν2∧μ). Let e∈H be a vector such that μe is a scalar spectral measure (for example a separating vector for von Neumann algebra generated by U – from the proof of Theorem 3.8). Since μ and μe are mutually absolutely continuous as scalar spectral measures for U, it follows that ν≪μe. By [7, Ch.IX, Lemma 8.6] there is a nonzero y∈[x]U,U∗ such that μy=ν. Therefore, y∈Hν1. On the other hand if y∈Hν2 then μy=ν≪ν2∧μ, which is a contradiction.

Corollary 3.11

Let ν1,ν2∈M+(T). If ν1≪ν2 then Hν1⊆Hν2.

Corollary 3.12

Let U be a unitary operator on H and μ be any scalar spectral measure for U. For ν1,ν2∈M+(T) the following are equivalent. Hν1=Hν2;

ν1∧μ and ν2∧μ are mutually absolutely continuous.

A matrix description of Cs(U)

Our first description of Cs(U) will involve matrix a representation of U with respect to an orthonormal basis. Though this might seem a bit difficult to apply at first, we will see, through our decomposition theorem from the previous section, that it can often yield a tangible description of Cs(U). Recall our notation from §2, where, for a fixed orthonormal basis B={uj}j⩾1 for a (separable) Hilbert space H and for A∈B(H), [A]B denotes the matrix representation of A with respect to B, and JB denotes the conjugation on H for which JBun=un for all n⩾1.

Theorem 4.1

Suppose U is a unitary operator on H. Then C∈Cs(U) if and only if there is a unitary operator V on H satisfying [V]Bt=[V]B;

[V]B[U]Bt=[U]B[V]B;

C=VJB.

Proof

Proposition 2.10 says that C is a conjugation on H if and only if C=VJB for some unitary V on H with [V]Bt=[V]B. Now, to fulfill the requirement that C∈Cs(U), observe thatCU∗C=U⟺(VJB)U∗(VJB)=U⟺V(JBU∗JB)(JBVJB)=U⟺[V]B[JBU∗JB]B[JBVJB]B=[U]B⟺[V]B[U]Bt[V∗]B=[U]B.

Notice the use of (2.12) above. Thus,CU∗C=U⟺[V]B[U]Bt=[U]B[V]B,

which completes the proof.

One can use the above discussion to describe Cs(U) when U is an n×n unitary matrix.

Theorem 4.2

Suppose U is an n×n unitary matrix with spectral decompositionU=Wξ1In1ξ2In2⋱ξdIndW∗,

where W is an n×n unitary matrix, ξ1,…,ξd∈T are the distinct eigenvalues of U, Inj is the nj×nj identity matrix, and n1+n2+⋯+nd=n. Then C∈Cs(U) if and only if4.3 C=WV1V2⋱VdJW∗,

where, for each 1⩽j⩽d, Vj is an nj×nj unitary matrix which satisfies Vjt=Vj, and J is the conjugation on Cn defined byJ[x1x2⋯xn]t=[x1¯x2¯⋯xn¯]t.

Proof

Suppose that C∈Cs(U). For the unitary matrixU~=ξ1In1ξ2In2⋱ξdInd,

Corollary 3.4 says that any C~∈Cs(U~) can be written asC~=C~|Eξ1⊕C~|Eξ2⊕⋯⊕C~|Eξd,

whereEj=ker(U~-ξjI)

(which is the span of appropriate standard basis vectors for Cn) andC~j:=C~|Eξ∈Cs(U~|Eξj)forall1⩽j⩽d.

Theorem 4.1 says that for each 1⩽j⩽d there is an nj×nj unitary matrix Vj such that such that C~j=VjJ|Ej and Vjt=Vj. Note that J|Ej fixes the appropriate standard basis vectors. Since U~|Eξj=ξjInj, condition (b) of Theorem 4.1 is automatic. Thus, every C~∈Cs(U~) takes the formV1V2⋱VdJ.

Now apply Lemma 2.3 and Proposition 2.6 to see that C takes the form in (4.3).

Conversely suppose that C is the conjugation from (4.3). Then, due to the fact that each of the unitary matrices Vj satisfy Vjt=Vj and JAJ=A¯ for any matrix A, CUC is equal toWV1V2⋱VdJξ1In1ξ2In2⋱ξdIndV1V2⋱VdJW∗=WV1V2⋱VdJξ1V1ξ2V2⋱ξdVdJW∗=WV1V2⋱Vdξ1¯V1¯ξ2¯V2¯⋱ξd¯Vd¯W∗=WV1V2⋱Vdξ1¯V1∗ξ2¯V2∗⋱ξd¯Vd∗W∗=Wξ1¯In1ξ2¯In2⋱ξd¯IddW∗=U∗.

Thus C∈Cc(U), which completes the proof.

In certain circumstances, we can extend Theorem 4.2 to the infinite dimensional setting. We mention two interesting examples of unitary operators from harmonic analysis.

Example 4.4

Let(Ff)(x)=12π∫-∞∞f(t)e-ixtdt,

denote the classical Fourier–Plancherel transform on L2(R) (defined initially on L1(R)∩L2(R) by the above integral and extended to be a unitary operator on L2(R)). It is known that σ(F)=σp(F)={1,-1,i,-i} and FHn=(-i)nHn, where Hn is the nth Hermite function, i.e.,Hn(x)=cne-x2/2hn(x),n⩾0,

cn is a constant for which ‖Hn‖L2(R)=1, and hn is the n-th Hermite polynomial. Moreover, B={Hn}n⩾0 forms an orthonormal basis for L2(R) [16, Ch.11]. Thus, using our earlier notation from Corollary 3.4,4.5 L2(R)=E1⨁E-i⨁E-1⨁Ei,

where B1={H4k}k⩾0, B-i={H4k+1}k⩾0, B-1={H4k+2}k⩾0, Bi={H4k+3}k⩾0, are orthonormal bases for E1,E-i,E-1,E-i respectively, we can write F in block operator form with respect to the decomposition in (4.5) asF=I|E1-iI|E-i-I|E-1iI|Ei.

Furthermore, any conjugation C on L2(R) for which CFC=F∗ must take the (block) formC=V1V-iV-1ViJ1J-iJ-1Ji,

where, for each k∈{1,-i,-1,i}, Jk is the conjugation on Ek which fixes every element of Bk and Vk is a unitary operator on Ek for which [Vk]Bkt=[Vk]Bk.

Example 4.6

Suppose(Hg)(x)=1πPV∫-∞∞g(t)x-tdt,

is the Hilbert transform on L2(R). Then σ(H)=σp(H)={i,-i} and, again using the notation from Corollary 3.4, L2(R)=Ei⨁E-i. Moreover, Ei has orthonormal basis Bi={fn}n⩾1, wherefn(x)=1π(x+i)n-1(x-i)n,

and E-i has orthonormal basis B-i={gn}n⩾0, wheregn(x)=1π(x-i)n(x+i)n+1.

See [16, Ch. 12] for the details. Moreover, by a similar discussion as in the previous example, any conjugation C on L2(R) for which CHC=H∗ must take the (block) formC=ViV-iJiJ-i,

where for each k∈{i,-i}, Jk is the conjugation on Ek which fixes every element of Bi and Vk is a unitary operator on Ek for which [Vk]Bkt=[Vk]Bk.

Natural conjugations on vector valued L2 spaces

This section provides a model for conjugations on vector valued Lebesgue spaces. It will be useful for our description of Cs(U) in §8 and also sets up our discussion of models for bilateral shifts, and their associated conjugations, in the next section.

For a Hilbert space H with norm ‖·‖H and μ∈M+(T), consider the set L0(μ,H) of all H-valued μ-measurable functions f on T and the Hilbert spaceL2(μ,H):={f∈L0(μ,H):‖f‖L2(μ.H):=(∫T‖f(ξ)‖H2dμ(ξ))1/2<∞}.

One often sees this using tensor notation as L2(μ)⊗H. Also consider L∞(μ,B(H)), the μ-essentially bounded B(H)-valued functions U on T. For U∈L∞(μ,B(H)), define the multiplication operator MU on L2(μ,H) by5.1 (MUf)(ξ)=U(ξ)f(ξ)

for f∈L2(μ,H) and μ-almost every ξ∈T. Clearly MU∈B(L2(μ,H)). If we use the notation U∗(ξ)=U(ξ)∗, one can verify thatMU∗=MU∗.

We let L∞(μ):=L∞(μ,C) to denote the set of all scalar valued μ-essentially bounded functions on T. For ease of notation, we will write Mφ, when φ∈L∞(μ), in place of the more cumbersome MφIH, that is,5.2 (Mφf)(ξ)=(MφIHf)(ξ)=φ(ξ)f(ξ)

for f∈L2(μ,H) and for μ-almost every ξ∈T. Of particular importance is when φ(ξ)=ξ which yields the (bilateral) shift Mξ on L2(μ,H). As a specific example, we have the bilateral shift (Mξf)(ξ)=ξf(ξ) on L2(m,T) when H=C.

Recall from §2 that AB(H) denotes the space of all bounded antilinear operators on H. By L∞(μ,AB(H)) we denote the space of all μ-essentially bounded and AB(H)-valued Borel functions on T. Analogously, as in (5.1), for C∈L∞(μ,AB(H)), define(ACf)(ξ)=C(ξ)f(ξ)

for f∈L2(μ,H) and for μ-almost every ξ∈T. One can check that AC∈AB(L2(μ,H)).

For any conjugation J on H, define the associated conjugation J on L2(μ,H) by5.3 (Jf)(ξ)=J(f(ξ)).

For example, when H=C and Jz=z¯ on C, then J is the conjugation on L2(μ) defined by Jf=f¯ and discussed earlier. One can check that for f∈L2(μ,H) and μ-almost every ξ∈T we have(JMξJf)(ξ)=(Mξ¯f)(ξ)

and thus,5.4 JMξJ=Mξ¯.

Proposition 5.6 below echos a result from [5, Proposition 4.2] and relies on the following (easily verified) lemma.

Lemma 5.5

Let U be a unitary operator and C be a conjugation on H. Then UC is a conjugation on H if and only if C∈Cs(U).

Proposition 5.6

Let J be a conjugation on H, J be defined by (5.3), and let U∈L∞(μ,B(H)) be such that U(ξ) is unitary for μ-almost every ξ∈T. Then the following hold. MUJ is a conjugation on L2(μ,H) if and only if U(ξ) is J–symmetric for μ-almost every ξ∈T.

When MUJ is a conjugation on L2(μ,H), we have (MUJ)Mξ(MUJ)=Mξ¯.

Proof

The proofs that the mapping MUJ is antilinear and isometric on the space L2(μ,H) are straightforward. Assume that MUJ is a conjugation. It follows from Lemma 5.5 that for every f∈L2(μ,H),(MUJf)(ξ)=U(ξ)(Jf)(ξ)=U(ξ)J(f(ξ))

must be equal to(JMU∗f)(ξ)=J((MU∗f)(ξ))=J((U∗(ξ)f)(ξ)).

Therefore, U(ξ)J=JU∗(ξ) for μ-almost every ξ∈T and thus, U(ξ) is J-symmetric for μ-almost every ξ∈T. The above argument can be reversed, which proves (a).

To prove (b), observe that for any f∈L2(μ,H) we use the fact that JU(ξ)J=U∗(ξ) for μ-almost every ξ∈T to see that(MUJ)Mξ(MUJf)(ξ)=U(ξ)(JMξMUJf)(ξ)=U(ξ)J((MξMUJf)(ξ))=U(ξ)J((ξMUJf)(ξ))=ξ¯U(ξ)J((MUJf)(ξ))=ξ¯U(ξ)J((JMU∗f)(ξ))=ξ¯U(ξ)U∗(ξ)f(ξ)=ξ¯f(ξ)=(Mξ¯f)(ξ).

□

Conjugations and bilateral shifts

Many interesting, and naturally occurring, unitary operators are bilateral shifts. Examples include the Hilbert and Fourier transforms on L2(R); the translation operator (Uf)(x)=f(x-1) on L2(R); the dilation operator (Uf)(x)=2f(2x) on L2(R); and the special class of multiplication operators Uf=ψf on L2(m,T) (where ψ is an inner function) which will be the focus of the next section. The fact that the above operators are bilateral shifts was carefully explained in [29]. This section gives an initial description of Cs(U) for this class of operators. Another description will be discussed in the next section. We begin with precisely what we mean by the term “bilateral shift”.

Definition 6.1

A unitary operator U on H is a bilateral shift if there is a subspace M⊆H for which UnM⊥M for all n∈Z\{0};

H=⨁n=-∞∞UnM.

In the above, note that U-1=U∗. The subspace M is called an associated wandering subspace for the bilateral shift U. Of course there is the bilateral shift Mξ on L2(m,T) defined by (Mξf)(ξ)=ξf(ξ) where the wandering subspace M is the space of constant functions.

Though the wandering subspace M in Definition 6.1 is not unique, its dimension is [26]. The term “bilateral shift ” comes from the fact that sinceH=⨁n=-∞∞UnM,

every x∈H can be uniquely represented asx=∑n=-∞∞Unxn,wherexn∈Mfor alln∈Z.

This allows us to define a natural unitary operator6.2 W:H→L2(m,M),W⨁n=-∞∞Unxn=∑n=-∞∞xnξn.

Moreover, WUW∗=Mξ, where Mξ is the bilateral shift from (5.2) defined on L2(m,M) by(Mξf)(ξ)=ξf(ξ),f(ξ)=∑n=-∞∞xnξn∈L2(m,M).

Example 6.2

Examples of bilateral shifts come from a variety of places. These operators were explored in [29] using a Wold-type decomposition. If U:L2(R)→L2(R) is the unitary translation operator defined by (Uf)(x)=f(x-1) and one sets M=χ[0,1]L2(R), then M satisfies conditions (a) and (b) of Definition 6.1 and hence U is unitarily equivalent to Mξ on L2(m,M).

Let U:L2(R)→L2(R) be the unitary dilation operator defined by (Uf)(x)=2f(2x). If ψ(x)=1for0⩽x<12,-1for12⩽x⩽1,0otherwise,

then (Haar) wavelet theory [6] says that the functions ψn,k(x):=2n2ψ(2nx-k),n,k∈Z,

form an orthonormal basis for L2(R). For fixed ℓ∈Z, define Wℓ:=⋁{ψℓ,k:k∈Z}.

Then Wℓ⊥Wℓ′ for all ℓ,ℓ′∈Z,ℓ≠ℓ′, and UWℓ=Wℓ+1 for all ℓ∈Z. This means that the subspace M=W0 satisfies conditions (a) and (b) of Definition 6.1 and hence U is unitarily equivalent to Mξ on L2(m,M).

For an inner function ψ (a bounded analytic function on D whose radial boundary values are unimodular for m-almost every ξ∈T), the operator Mψf=ψf is unitary on L2=L2(m,T) and the model space M=Kψ=H2∩(ψH2)⊥ satisfies the hypothesis of conditions (a) and (b) of Definition 6.1 [29, Proposition 5.17]. Hence Mψ is unitarily equivalent to Mξ on L2(m,M). Observe that if U is any bilateral shift on H with wandering subspace N with dimN=N∈N∪{∞}, then the above discussion shows that U is unitarily equivalent to Mψ on L2, where ψ is any inner function whose degree is N. When the inner function ψ is a finite Blaschke product with N zeros (repeated according to multiplicity), the degree of ψ is defined to be N. In all other cases, the degree of ψ is defined to be N=∞. In the next section, will give a concrete description of Cs(Mψ) written in terms of operators on L2.

For a bilateral shift U on H we wish to describe Cs(U). Since WUW∗=Mξ on L2(m,M), where W:H→L2(m,M) is the unitary operator from (6.2), Lemma 2.3 says that C=WCW∗ is a conjugation on L2(m,M) such that CMξC=Mξ¯. The following result from [5, Thm. 4.8] describes C.

Theorem 6.4

For a conjugation C on L2(m,M), the following are equivalent. C∈Cs(Mξ);

There is a C∈L∞(m,AB(M)) such that C=AC and C(ξ) is a conjugation for almost all ξ∈T;

For any conjugation J on M there is a U∈L∞(m,B(M)) such that U(ξ) is a J–symmetric unitary operator for almost all ξ∈T and C=MUJ.

This yields a description of Cs(U) when U is a bilateral shift. Recall the unitary operator W:H→L2(m,M) from (6.2) associated with a bilateral shift with wandering subspace M.

Corollary 6.5

Let U be a bilateral shift on H with an associated wandering subspace M. For a conjugation C on H, the following are equivalent: C∈Cs(U);

C=WCW∗ is a conjugation on L2(m,M) that satisfies any of the equivalent conditions of Theorem 6.4.

Example 6.6

For the bilateral shift Mξ on L2=L2(m,T), we can examine the set Cs(Mξ). Here M=C (the constant functions) and J:C→C is Jz=z¯ and so J:L2→L2 is Jf=f¯. If u∈L∞ is unimodular on T, then Muf=uf is unitary and JMuJ=Mu¯. Theorem 6.4 says that any C∈Cs(Mξ) must take the form C=MuJ for some unimodular u∈L∞. We will see another path to this result in Example 7.10.

Example 6.7

In a similar way as with the previous example, let B={vk}k⩾1 be an orthonormal basis for M. Here N=dimM (which might be infinite). Recall the conjugation JB on M defined by JBvk=vk for all k⩾1 (see (2.9)). Suppose U∈L∞(m,B(M)) is such that U(ξ) is unitary for almost every ξ∈T. One can check that [JU(ξ)J]B=[U(ξ)]B¯ for almost every ξ∈T. Thus, for JU(ξ)J=U(ξ)∗ it must be the case that [U(ξ)∗]B=[U(ξ)]B¯. This means that every conjugation C on L2(m,M) for which CMξC=Mξ¯ must take the form (Cf)(ξ)=U(ξ)J(f(ξ)), where [U(ξ)∗]B=[U(ξ)]B¯ for almost every ξ∈T. Note that for any conjugation J on M, we can find an orthonormal basis B for the M for which J=JB [18, Lemma 1]. Thus, in a way, the example above is canonical.

The results from [29, Theorem 3.3] show that for any unitary operator U on H we have H=K⊕K′, where K and K′ reducing subspaces for U, where U|K is a bilateral shift and U|K′ has no a bilateral shift part. For a given conjugation C, even assuming that U is C-symmetric, the conjugation C cannot be nesseserily decomposed according to this decomposition.

Example 6.8

Consider the same operator as in [29, Example 5.6]. Namely, let H=L2(Ω1)⊕L2(Ω2)⊕L2(Ω1), where Ω1, Ω2 are (Lebesgue) measurable disjoint subsets of T such that Ω1∪Ω2=T andU(f,g,h)(z)=(ξf(ξ),ξg(ξ),ξh(ξ))forf,h∈L2(Ω1),g∈L2(Ω2).

Define the conjugationC(f,g,h)(ξ)=(h(ξ)¯,g(ξ)¯,f(ξ)¯)forf,h∈L2(Ω1),g∈L2(Ω2)

and observe that CUC=U∗. As in [29, Example 5.6], the decomposition H=K⊕L2(Ω1) with K=L2(T)=L2(Ω1)⊕L2(Ω2), is a desired bilateral shift decomposition, but it is not C invariant (CK⊈K).

Unitary multiplication operators on L2(m,T)

As mentioned in Example 6.2(c), we have a model for any bilateral shift U on H as the multiplication operator Mψ on L2=L2(m,T), where ψ is an inner function whose degree is that of the (uniquely defined) dimension of any wandering subspace for U. In this section we give a concrete and tangible description of Cs(Mψ). If J is the conjugation Jf=f¯ on L2 and C∈Cs(Mψ), then A=CJ is a unitary operator on L2 for which AMψ=MψA. Thus, in order to describe Cs(Mψ), we first need to characterize the bounded operators on L2 that commute with Mψ.

Remark 7.1

For an inner function ψ, a known result (see for example [29, Proposition 5.17]), says that7.2 L2=⨁n=-∞∞ψnKψ,

where Kψ=H2⊖ψH2 is the model space associated with ψ. In other words, Kψ is a wandering subspace for Mψ (recall Definition 6.1). Let us set up some notation to be used below. Let N=dimKψ∈N∪{∞}. Observe that N is finite if and only if ψ is a finite Blaschke product with N zeros, repeated according to multiplicity [15, Prop. 5.19]. Also define the space⨁1⩽j⩽NL2=L2⊕L2⊕⋯⊕L2.

The norm of an element f=[fj]1⩽j⩽Nt∈⨁1⩽j⩽NL2 is‖f‖=∑1⩽j⩽N‖fj‖221/2.

When N=∞, we need to assume that the sum defining ‖f‖ is finite. Furthermore, the operator ⨁1⩽j⩽NMξ (called the inflation of the bilateral shift Mξ on L2) is given by⨁1⩽j⩽NMξf(ξ)=ξf(ξ)=[ξfj(ξ)]1⩽j⩽Nt.

We also defineℓN2:=x=[xj]1⩽j⩽Nt,xj∈C:‖x‖ℓN2=∑1⩽j⩽N|xj|21/2<∞.

When N=∞, note that ℓN2 is equal to ℓ+2 which was the space discussed earlier in (2.8). Finally, observe that7.3 ⨁1⩽j⩽NMξ≅Mξ|L2(m,ℓN2).

Theorem 7.4

For an inner function ψ, let {hj}1⩽j⩽N be an orthonormal basis for the model space Kψ. Then we have the following. Every f∈L2 has the unique decomposition 7.5 f=∑1⩽j⩽Nhj·(fj∘ψ),

where each fj belongs to L2 and ‖f‖=(∑1⩽j⩽N‖fj‖2)12.

The operator W:L2→⨁1⩽j⩽NL2,Wf=[f1f2f3…]t,

is unitary and W∗[kj]1⩽j⩽Nt=∑1⩽j⩽Nhj·(kj∘ψ).

WMψW∗=⨁1⩽j⩽NMξ and WMψ¯W∗=⨁1⩽j⩽NMξ¯.

If A∈B(L2) satisfies AMψ=MψA, then Af=∑1⩽j⩽N(fj∘ψ)∑1⩽k⩽Nhk·(φkj∘ψ),

where Φ=[φij]1⩽i,j⩽N∈L∞(m,B(ℓN2)).

If Φ=[φij]1⩽i,j⩽N∈L∞(m,B(ℓN2)), then the operator A defined in (d) is unitarily equivalent to MΦ on ⨁1⩽j⩽NL2, i.e., [fj]1⩽j⩽Nt↦Φ[fj]1⩽j⩽Nt.

Proof

By (7.2), every f∈L2 can be written (uniquely) as7.6 f=∑j=-∞∞ψj∑1⩽k⩽Najkhk.

Moreover, since ψjKψ⊥ψ′Kψ for all j≠j′ and {hk}1⩽k⩽N forms an orthonormal basis for Kψ, we have7.7 ‖f‖2=∑j=-∞∞∑1⩽k⩽N|ajk|2<∞.

Rewrite the expression in (7.6) asf=∑1⩽k⩽Nhk∑j=-∞∞ajkψj.

By (7.7) the function fk=∑j=-∞∞ajkzj belongs to L2 and thus,f=∑1⩽k⩽Nhk·(fk∘ψ).

Finally, by Parseval’s theorem and (7.7), note that∑1⩽k⩽N‖fk‖2=∑1⩽k⩽N∑j=-∞∞|ajk|2=‖f‖2.

This verifies statements (a) and (b). To verify statement (c), note thatWMψW∗[kj]1⩽j⩽Nt=WMψ∑1⩽j⩽Nhj·(kj∘ψ)=W∑1⩽j⩽Nhj·ψ·(kj∘ψ)=W∑1⩽j⩽Nhj·((ξkj)∘ψ)=[ξkj(ξ)]1⩽j⩽Nt.

This shows that WMψW∗=⨁1⩽j⩽NMξ. Similarly, one can verify the second equality in (c).

To prove (d) and (e), recall from [33, Chapter III] that the bounded operators on the space ⨁1⩽j⩽NL2 that commute with ⨁1⩽j⩽NMξ must take the form of multiplication by the matrix function Φ=[φij]1⩽i,j⩽N∈L∞(m,ℓN2). Putting this all together, we see that if A∈B(L2) commutes with Mψ, then WAW∗ commutes with ⨁1⩽j⩽NMξ and so WAW∗=MΦ, equivalently A=W∗MΦW. This translates to an operator on L2 byAf=W∗ΦWf=W∗[φij]1⩽i,j⩽N[f1f2…]t=W∗∑1⩽j⩽Nφ1jfj∑1⩽j⩽Nφ2jfj∑1⩽j⩽Nφ3jfj…t=h1·∑1⩽j⩽Nφ1jfj∘ψ+h2·∑1⩽j⩽Nφ2jfj∘ψ+…=(f1∘ψ)·∑1⩽k⩽Nhk·(φk1∘ψ)+(f2∘ψ)·∑1⩽k⩽Nhk·(φk2∘ψ)+⋯,

which completes the proof of (d) and (e).

From our earlier discussion, we know that the standard conjugation Jf=f¯ on L2 induces the standard conjugation on J on ⨁1⩽j⩽NL2 byJF=[f¯1f¯2…]t=F¯

Here is our description of Cs(Mψ) when ψ is inner.

Theorem 7.8

Suppose that ψ is an inner function and {hj}1⩽j⩽N is an orthonormal basis for Kψ. Then C∈Cs(Mψ) if and only if there is a Φ=[φij]1⩽i,j⩽N∈L∞(m,ℓN2) such that Φ∗Φ=I almost everywhere on T;

Φt=Φ almost everywhere on T;

Cf=∑1⩽j⩽N(f¯j∘ψ)∑1⩽k⩽Nhk·(φk,j∘ψ), for all f∈L2 with the decomposition from (7.5).

Proof

Suppose C∈Cs(Mψ). Note that C~:=WCW∗ defines a conjugation on ⨁1⩽j⩽NL2. Since CMψC=Mψ¯, we can use Theorem 7.4(c) to see thatC~⨁1⩽j⩽NMξC~=⨁1⩽j⩽NMξ¯.

Since ⨁1⩽j⩽NMξ on ⨁1⩽j⩽NL2 is unitary equivalent to Mξ on L2(m,ℓN2) (recall (7.3)), Theorem 6.4 yields (with the choice of conjugation J on ℓN2 being Jx=x¯), a matrix-valued function Φ=[φij]1⩽i,j⩽N∈L∞(m,ℓN2) that is unitary valued m-almost everywhere, i.e., ΦΦ∗=Φ∗Φ=I, and such that C~=MΦJ. Moreover, since C~2=I we see thatI=(MΦJ)(MΦJ)=MΦ(JMΦJ)=MΦMΦ¯

and so Φ¯=Φ∗. Combine this with the previous identity to see that Φt=Φ. So far we have shown that if C∈Cs(Mψ) then C=W∗(MΦJ)W where Φ satisfies the conditions of (a) and (b). If Φ satisfies the conditions (a) and (b) then a short argument will show that Φ is unitary valued almost everywhere and the condition (b) will show that JΦJ=Φ∗ almost everywhere. Proposition 5.6 now says that MΦJ∈Cs(Mξ) and hence W∗(MΦJ)W∈Cs(Mψ). So we have shown that C∈Cs(Mψ) if and only if C=W∗(MΦJ)W, where Φ satisfies the conditions in (a) and (b).

It remains to verify the formula in (c). Observe that for all f=∑1⩽j⩽Nhj·(fj∘ψ)∈L2,Cf=W∗MΦJWf=W∗[φij]1⩽i,j⩽N[f¯1f¯2…]t=W∗∑1⩽j⩽Nφ1jf¯j∑1⩽j⩽Nφ2jf¯j∑1⩽j⩽Nφ3jf¯j…t=h1·∑1⩽j⩽Nφ1jf¯j∘ψ+h2·∑1⩽j⩽Nφ2jf¯j∘ψ+…=(f¯1∘ψ)·∑1⩽k⩽Nhk·(φk1∘ψ)+(f¯2∘ψ)·∑1⩽k⩽Nhk·(φk2∘ψ)+⋯.

Conversely, suppose there is a Φ=[φij]1⩽i,j⩽N∈L∞(m,ℓN2) such that conditions (a), (b), and (c) hold. Theorem 7.4(e) and the argument above shows that C∈Cs(Mψ).

Remark 7.9

The description of Cs(Mψ) for an inner function ψ depends on knowing an orthonormal basis for the model space Kψ. There are several “natural” bases one could choose. See [15, Prop. 5.25] for some particular examples when dimKψ is finite.

Example 7.10

Suppose ψ(z)=z. In this case, Mψ becomes the bilateral shift Mξ on L2. Furthermore, Kψ=C (the constant functions) and the expansion of an f∈L2 from Theorem 7.4 becomes the classical Fourier expansion. Finally, Theorem 7.4 says that every C∈Cs(Mξ) must take the form (Cf)(ξ)=u(ξ)f(ξ)¯ for some u∈L∞ that is unimodular almost everywhere. We observed this by a different method in Example 6.6.

Example 7.11

When ψ(z)=z2 above, one can check that the functions h1(z)≡1 and h2(z)=z form an orthonormal basis for Kψ. Furthermore, using the notation from Theorem 7.4,7.12 f1(ξ)=∑j=-∞∞f^(2j)ξjandf2(ξ)=∑j=-∞∞f^(2j+1)ξj.

Then, one can check thatf(ξ)=h1(ξ)f1(ξ2)+h2(ξ)f2(ξ2)=f1(ξ2)+ξf2(ξ2).

Theorem 7.8 says that every C∈Cs(Mξ2) takes the form(Cf)(ξ)=f1(ξ2)¯(φ11(ξ2)+ξφ21(ξ2))+f2(ξ2)¯(φ21(ξ2)+ξφ22(ξ2)),

where φij, 1⩽i,j⩽2, are bounded measurable functions on T for which φ12(ξ)=φ21(ξ) and7.13 φ11(ξ)¯φ21(ξ)¯φ21(ξ)¯φ22(ξ)¯φ11(ξ)φ21(ξ)φ21(ξ)φ22(ξ)=1001

for almost every ξ∈T. The condition from (7.13) is equivalent to the identities|φ11(ξ)|2+|φ21(ξ)|2=1,|φ11(ξ)|=|φ22(ξ)|,φ11(ξ)¯φ21(ξ)+φ21(ξ)¯φ22(ξ)=0

for almost every ξ∈T.

To make this more tangible, let t=Arg(ξ)∈(-π,π] and let s(t),α(t), β(t), and γ(t) be any 2π–periodic bounded real-valued (Lebesgue) measurable functions. Setφ11(ξ)=eiα(t)s(t),φ22(ξ)=eiβ(t)s(t),φ21(ξ)=φ12(ξ)=eiγ(t)1-s2(t),

and observe that the conditions above yieldγ(t)=12(π+α(t)+β(t)),

which shows that0⩽s(t)⩽1,φ11(ξ)=eiα(t)s(t),φ22(ξ)=eiβ(t)s(t),φ21(ξ)=φ12(ξ)=iei2(α(t)+β(t))1-s2(t).

Using the notation from (7.12), this means that every C∈Cs(Mξ2) must take the form(Cf)(ξ)=f1(ξ2)¯(eiα(2t)s(2t)+iξei2(α(2t)+β(2t))1-s2(2t))+f2(ξ2)¯(iei2(α(2t)+β(2t))1-s2(2t)+ξeiβ(2t)s(2t)),

where t=Arg(ξ)∈(-π,π] and s(t),α(t),β(t) are any 2π–periodic bounded real-valued measurable functions.

As a specific nontrivial example we can have the interesting C∈Cs(Mξ2) defined by(Cf)(ξ)=f1(ξ2)¯(sin(2t)+ξcos(2t))+f2(ξ2)¯(cos(2t)-ξsin(2t)),

where t=Arg(ξ) (s(t)=sint, α(t)≡0, β(t)≡-π)

Another interesting example comes from setting s(t)≡s, α(t)=λt, β(t)=-π-λt, λ∈R, which yields(Cf)(ξ)=f1(ξ2)¯(seiλt+ξ1-s2)+f2(ξ)¯(1-s2+iξse-iλt).

Conjugations via the spectral theorem

This section describes Cs(U) using the multiplicity version of the spectral theorem. Important applications of this are Theorem 8.6 and Theorem 8.8 below which connect the invariant subspaces of C∈Cs(U) with the hyperinvariant subspaces of U.

We begin with the following (multiplicity) version of the spectral theorem for unitary operators from [7, Ch. IX, Theorem 10.20]. The reader might need a refresher of the notation from §5.

Theorem 8.1

(Spectral Theorem) For a unitary operator U on a separable Hilbert space H, there are mutually singular measures μ∞,μ1,μ2,…∈M+(T), along with Hilbert spaces H∞,H1,H2,… each with corresponding dimHk=k, k=∞,1,2,3,⋯, and a unitary operatorI:H→L2(μ∞,H∞)⊕L2(μ1,H1)⊕L2(μ2,H2)⊕⋯

such that IUI∗ is equal to the unitary operatorMξ(∞)⊕Mξ(1)⊕Mξ(2)⊕⋯,

where for i=∞,1,2,3,…,Mξ(i):L2(μi,Hi)→L2(μi,Hi),Mξ(i)f(ξ)=ξf(ξ).

The main driver of the results of this section is the following.

Theorem 8.2

Let U be a unitary operator on a separable Hilbert space H with spectral representation as in Theorem 8.1. For a conjugation C on H, the following are equivalent. C∈Cs(U),

For each j=∞,1,2,⋯ there are conjugations Cj on L2(μj,Hj) such that Cj∈Cs(Mξ(i)) and C=I∗(⨁Cj)I.

For each j=∞,1,2,⋯ there are Cj∈L∞(μj,AB(Hj)) such that Cj(ξ) is a conjugation for μj almost every ξ∈T and C=I∗(⨁ACj)I.

For each i=∞,1,2,⋯ and for any conjugation Ji on Hi there is a Ui∈L∞(μi,B(Hi)) such that Ui(ξ) is unitary and Ji–symmetric for μi-almost every ξ∈T and C=I∗(⨁UiJi)I=I∗(⨁Ui)(⨁Ji)I.

Proof

(a)⟹(d): Set8.3 M~ξ:=IUI∗=Mξ(∞)⊕Mξ(1)⊕Mξ(2)⊕⋯,

and, for a conjugation C on H define C~=ICI∗ which, by Lemma 2.3, will be a conjugation onLH2:=L2(μ∞,H∞)⊕L2(μ1,H1)⊕L2(μ2,H2)⊕⋯.

Assuming that CUC=U∗, we see that C~M~ξC~=M~ξ¯.

Let Ji be any a conjugation on Hi. Then, as in (5.3), this induces a conjugation Ji on L2(μi,Hi) defined by(Jifi)(ξ)=Ji(fi(ξ)),fi∈L2(μi,Hi).

Define a conjugation J~ on LH2 byJ~:=J∞⊕J1⊕J2⋯.

From (5.3) we see that J~M~ξJ~=M~ξ¯. Now observe thatM~ξ(C~J~)=(C~M~ξ¯)J~=(C~J~)M~ξ.

This says that the operator C~J~ commutes with Mξ~. The spectral theorem applied to Mξ~ also yields the commutant [7, p.  307, Theorem 10.20], namely there are operator valued functions Ui∈L∞(μi,B(Hi)) for i=∞,1,2,⋯ such thatC~J~=MU∞⊕MU1⊕MU2⊕⋯.

Since C~J~ is unitary (being linear, isometric, and onto), it follows that each M~Ui is unitary and, consequently, Ui(ξ) is unitary for μi-almost every ξ∈T. Therefore,C~=(⨁MUi)(⨁Ji)=⨁(MUiJi).

Since each C~|L2(μi,Hi) is a conjugation, Proposition 5.6 says that the operator Ui(ξ) is Ji–symmetric μi almost everywhere. Thus, we have verified the implication (a)⟹(d).

(d)⟹(c): For each i=∞,1,2,…, take Ci=UiJi.

(c)⇒(b): For each i=∞,1,2,…, it is enough to take Ci=ACi and prove thatACiMξ(i)=Mξ¯(i)ACi.

Indeed, for each fi∈L2(μi,Hi) we have(ACiMξ(i)fi)(ξ)=Ci(ξ)(Mξ(i)fi)(ξ)=Ci(ξ)(ξfi(ξ))=ξ¯Ci(ξ)fi(ξ)=ξ¯(ACifi)(ξ)=(Mξ¯(i)ACifi)(ξ).

(b)⟹(a): Recalling the notation from (8.3), note thatCUC=I(⨁Ci)(⨁Mξ(i))(⨁Ci)I∗=I(⨁CiMξ(i)Ci)I∗=I(⨁Mξ¯(i))I∗=IMξ¯~I∗=U∗.

□

The multiplication operator Mξ on L2(μ,H) is a special case of Theorem 8.2 – which we record here for what follows.

Theorem 8.4

Let μ∈M+(T) and C be a conjugation on L2(μ,H). Then following are equivalent. C∈Cs(Mξ).

There are C∈L∞(μ,AB(H)) such that C(ξ) is a conjugation for μ almost every ξ∈T and C=AC.

For any conjugation J on H there is a U∈L∞(μ,B(H)) such that U(ξ) is unitary and J–symmetric for μ almost every ξ∈T and C=MUJ.

As an application of the above, we have the following connection between conjugations and hyperinvariant subspaces.

Proposition 8.5

Let μ∈M+(T). If K⊆L2(μ,H) is an invariant subspace for every C∈Cs(Mξ), then K is hyperinvariant for Mξ.

Proof

For any fixed conjugation J on H, (5.4) says that J∈Cs(Mξ) and thus K is invariant for J. By Theorem 8.4, K is also invariant for all of the conjugations MuJ, where u∈L∞(μ) is unimodular μ-almost everywhere. Therefore, K is invariant for every Mu=MuJJ, where u∈L∞(μ) is unimodular. From here, one can argue that K is also invariant for MχΩ for any Borel set Ω⊆σ(U) (indeed let u=1 on Ω and -1 on T\Ω and note that χΩ=(1+u)/2) and, consequently, for any Mv where v∈L∞(μ).

Now fix any unitary U0 on H and let U0≡U0∈L∞(μ,B(H)) denote the operator-valued constant function. By our discussion in the introduction (see also Proposition 2.5), U0 is J0–symmetric for some conjugation J0. Let J0 be the conjugation on L(μ,H) given by (J0f)(ξ)=J0(f(ξ)). Therefore, K is invariant for J0 and for U0J0 and thus for U0. Similarly, K is invariant for U0∗. It follows that K is invariant for the von Neumann algebra containing all constant unitary valued functions in B(H). Finally, K is invariant for the von Neumann algebra generated by {Mv:v∈L∞(μ)} and the constant operator-valued functions from L∞(μ,B(H)). From here, one can fashion an argument that K is invariant for every element of L∞(μ,B(H)). Since L∞(μ,B(H)) forms the class of operators MΦ, Φ∈L∞(μ,B(H)), that commute with Mξ, K is hyperinvariant for Mξ.

From here, we can state our main connection between conjugations and hyperinvariant subspaces.

Theorem 8.6

Let U be a unitary operator on a separable Hilbert space H. If M⊆H is an invariant subspace for every C∈Cs(U), then M is hyperinvariant for U.

Proof

The notation from Theorem 8.2 proves that the invariance of M for all C∈Cs(U) implies that IM=⨁Ki, where the subspace Ki is invariant for all all MUiJi, where Ui∈L∞(μi,Hi) is unitary valued μ-almost everywhere. Now apply Proposition 8.5 to see that Ki is hyperinvariant for each Mξ(i). Finally, using the description of the commutant of Mξ [7, p. 307, Theorem IX.10.20], we obtain that IM is hyperinvariant for Mξ. This shows that M is a hyperinvariant subspace for U.

The simple example bellow illustrates why requirement in Theorem 8.6 that M is invariant for every C∈Cs(U) is an important one.

Example 8.7

Consider the diagonal unitary operator U∈B(C2) defined by U=diag[λ,λ] (where λ≠0) with respect to the standard basis {e1,e2} for C2. The only hyperinvariant subspaces for U are the trivial ones {0} and C2 (since any 2×2 matrix commutes with U). Define two conjugations C1, C2 on C2 byC1(e1)=e1,C1(e2)=e2andC2(e1)=e2,C2(e2)=e1

(and of course extend antilinearity to all of C2). One can check that C1,C2∈Cs(U). Note that each Ci separately has more invariant subspaces than the trivial ones {0} and C2, but only the trivial subspaces are simultaneously invariant for both of them.

Our summary theorem connecting hyperinvariant subspaces and conjugations is the following.

Theorem 8.8

Let U be a unitary operator on a separable Hilbert space H with spectral measure E(·). For a subspace M⊆H, the following are equivalent. M is hyperinvariant;

M=E(Ω)H for some Borel set Ω⊆σ(U);

M=Hμ for some μ∈M+(T);

CM⊆M for every C∈Cs(U).

Proof

(a)⟹(b) is Theorem 3.7. (b)⟹(c) is Theorem 3.8. (c)⟹(d) is Theorem 3.2. (d)⟹(a) is Theorem 8.6.

Remark 8.9

One can use Theorem 8.2 to give an alternate proof of Theorem 4.2 (the description of Cs(U) when U is an n×n unitary matrix). The same is true for Examples 4.4 and 4.6.

This work was supported by the NSERC Discovery Grant (Canada), the Fullbright Foundation, the Canada Research Chair program, and by the Ministry of Science and Higher Education of the Republic of Poland.

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