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

920
10.1007/s13324-024-00920-3
Article
Conjugations of unitary operators, II
Mashreghi Javad 1
Ptak Marek 2
Ross William T. wross@richmond.edu

3
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
10 5 2024
10 5 2024
2024
14 3 5622 2 2024
16 4 2024
17 4 2024
© The Author(s) 2024
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For a unitary operator U on a separable complex Hilbert space H, we describe the set Cc(U) of all conjugations C (antilinear, isometric, and involutive maps) on H for which CUC=U. As this set might be empty, we also show that Cc(U)≠∅ if and only if U is unitarily equivalent to U∗.

Keywords

Unitary operators
Conjugations
Model spaces
Shift operators
Invariant subspaces
Mathematics Subject Classification

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

This is the second in a series of two papers that explore conjugations of unitary operators on Hilbert spaces. The first paper [19] explored, for a given unitary operator U on a Hilbert space H, the antilinear, isometric, and involutive maps C on H, i.e., conjugations, for which CUC=U∗. An argument with the spectral theorem says there will always be a conjugation C with this property. Moreover, [19] contains various characterizations of the set of all such conjugations C for which CUC=U∗. These conjugations are the “symmetric conjugations” for U.

The purpose of this paper is to explore, for a given unitary U on H, the set1.1 Cc(U):={CisaconjugationonH:CUC=U}.

These are known as the “commuting conjugations” for U [3, 4]. The subscript c in the definition of Cc(U) might initially seem superfluous but we will use it anyway to distinguish this set from Cs(U) (notice the s in the subscript), the “symmetric conjugations” mentioned in the previous paragraph. For an easy example of a commuting conjugation, consider the unitary operator (Uf)(ξ)=ξf(ξ), the bilateral shift on L2(m,T), where m is normalized Lebesgue measure on the unit circle T. One can check that the map (Jf)(ξ)=f(ξ¯)¯ on L2(m,T) defines a conjugation which satisfies JUJ=U. Moreover (see Example 6.16), any conjugation C on L2(m,T) for which CUC=U takes the form (Cf)(ξ)=u(ξ)(Jf)(ξ), where u∈L∞(m,T) is both unimodular and satisfies u(ξ)=u(ξ¯) a.e.  on T. An analogous result holds when (Uf)(ξ)=ξf(ξ) on the vector-valued Lebesgue space L2(m,H), but not always on L2(μ,H) for a general positive measure μ on T (see Sect. 4 and the discussion below).

The first issue one needs to resolve is whether, for a given unitary operator U on H, there are any conjugations C for which CUC=U. Indeed, using the known fact from [16] (see also Proposition 2.8 below) that any unitary operator can be written as a composition of two conjugations, one can fashion a quick argument (see Lemma 2.9) to see that if Cc(U)≠∅, then U≅U∗ (i.e., U is unitarily equivalent to its adjoint U∗). One of the main results of this paper (Corollary 5.4) is the converse.

Theorem 1.2

For a unitary operator U on a complex separable Hilbert space H, the following are equivalent. Cc(U)≠∅;

U≅U∗.

Notice how condition (b) in Theorem 1.2 places some restrictions on the class of unitary operators which have commuting conjugations in that, at the very least, the spectrum σ(U) of U must be symmetric with respect to the real axis.

We give concrete descriptions of Cc(U) for many classes of unitary operators U such as the bilateral shift on L2(m), the related bilateral shift on L2(μ), the bilateral shift on the vector-valued L2(μ,H), multiplication by an inner function on L2(m), general bilateral shifts, the Fourier transform, and the Hilbert transform. The main driver all of these results comes from the classical spectral theorem for unitary operators (multiplicity version) [5, p.  307, Ch. IX, Theorem 10.20] which says that any unitary operator U on H is unitarily equivalent to1.3 M~:=Mξ(∞)⊕Mξ(1)⊕Mξ(2)⊕⋯,

onLH2:=L2(μ∞,H∞)⊕L2(μ1,H1)⊕L2(μ2,H2)⊕⋯

where for i=∞,1,2,3,…, μi are finite positive Borel measures on T,Mξ(i):L2(μi,Hi)→L2(μi,Hi),(Mξ(i)f)(ξ)=ξf(ξ),

and Hi are i-dimensional Hilbert spaces. Let I:H→LH2 denote the isometric isomorphism that induces the unitary equivalence of U and M~. In Lemma 5.2 we show that if Cc(U)≠∅, then μic≪μi for all i=∞,1,2,⋯. Here μic(Ω):=μi(Ω∗), where Ω∗:={ξ¯:ξ∈Ω}. Using these tools, the main description of Cc(U) is the following. We refer the reader to Sect. 4 for the precise definitions of the parameters J# and U#.

Theorem 1.4

Let U be a unitary operator and C be a conjugation on H. With the notation above, assuming that μic≪μi for i=∞,1,2,⋯, the following are equivalent C∈Cc(U);

For each i=∞,1,2,⋯, there are conjugations Ci on L2(μi,Hi) such that Mξ(i) is Ci–commuting and C=I∗(⨁Ci)I;

For each i=∞,1,2,⋯ and any conjugation J(i) on Hi, there is a unitary operator-valued function U(i)∈L∞(μi,B(Hi)) such that J(i)U(i)(ξ)J(i)=U(i)(ξ)# for μi-a.e.  ξ∈T and C=I∗(⨁U(i)J#(i))I=I∗(⨁U(i))(⨁J#(i))I.

This theorem is stated and proven in Sect. 5 and the concrete characterizations of Cc(U), for particular classes of unitary operators, will be given in Sect. 6. Parallel to a discussion in the first paper [19] in this series, we discuss the (closed) subspaces K of H for which CK⊆K for all C∈Cc(U) in Sect. 7.

Basics facts about conjugations

All Hilbert spaces H in this paper are separable and complex. Let B(H) denote the space of all bounded linear transformations on H and AB(H) denote the space 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 it satisfies the additional conditions that ‖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 [8, 9, 11–13]. More recently, conjugations were explored in [3, 4, 6, 7, 18, 21].

Example 2.2

Many types of conjugations were outlined in [11–13]. Below are a few basic ones that are relevant to this paper. The mapping Cf=f¯ defines a conjugation on a standard Lebesgue space L2(μ,X). In particular, the mapping Cx=x¯ defines a conjugation on Euclidean space Cn. Throughout this paper we will use the superscript t to represent the transpose of a matrix. In addition, vectors x in Cn will be viewed as column vectors since, for an n×n matrix A of complex numbers, we will often consider linear transformations on Cn defined by x↦Ax.

The mapping (Cf)(ξ)=f(ξ¯)¯ defines a conjugation on L2(μ,T) for any finite positive Borel measure on T.

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

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.

Example 2.4

We have already discussed the how the mapping (Cf)(ξ)=f(ξ¯)¯ on L2(m,T) is a conjugation that commutes with the bilateral shift (Uf)(ξ)=ξf(ξ). Here are a few other examples. The conjugation (Cf)(x)=f(x)¯ on L2(R) commutes with the unitary operator (Uf)(x)=f(x-1). This conjugation also commutes with the Hilbert transform.

The conjugation (Cf)(x)=f(-x)¯ on L2(R) commutes with the Fourier–Plancherel transform.

Recalling the definition of Cc(U) from (1.1), let us make a few elementary observations. One can argue from (2.1) that2.5 Cc(U)=Cc(U∗).

Commuting conjugations are stable under unitary equivalence.

Proposition 2.6

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

If U is unitary and C is a conjugation on H, then UC∈AB(H) and is isometric. This next result has a straightforward proof and determines when UC is involutive and hence a conjugation.

Lemma 2.7

Let U be a unitary operator and C be a conjugation on H. Then UC is a conjugation if and only if CUC=U∗.

We recall the following result from [16] (see also Proposition 2.5 from [19]) which shows that any unitary operator can be built from conjugations.

Proposition 2.8

Let U be a unitary operator on H. Then there are conjugations J1 and J2 on H such that U=J1J2. Moreover, J1UJ1=U∗ and J2UJ2=U∗.

In the introduction we showed that although every unitary operator U satisfies CUC=U∗ with respect to some conjugation C, it is possible for Cc(U) (the commuting conjugations for U) to be the empty set. Below we begin to determine when this happens (and bring this discussion to fruition in Corollary 5.4).

Lemma 2.9

If U is a unitary operator on H and Cc(U)≠∅, then U≅U∗.

Proof

Let J1 be as in Proposition 2.8, C∈Cc(U), and define V=J1C. Clearly V is unitary (since it is linear, isometric, and onto) and VU=J1CU=J1UC=U∗J1C=U∗V. Thus, U≅U∗. □

Conjugations and spectral measures

A version of the spectral theorem for unitary operators [5, Ch. IX, Thm. 2.2] (see also [17]) says that if U is a unitary operator on H, then there is a unique spectral measure E(·) on T such that3.1 U=∫ξdE(ξ).

Moreover, for any spectral measure E(·) on T, there is a unique unitary operator U associated with E(·) via (3.1).

For a spectral measure E(·) and x,y∈H, the functionμx,y(·):=⟨E(·)x,y⟩

defines a finite complex Borel measure on T and for each x∈H,μx:=μx,x

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

For a complex Borel measure μ on T, define a new Borel measure μc on the Borel subsets Ω of T by3.2 μc(Ω):=μ(Ω∗),whereΩ∗:={ξ¯:ξ∈Ω}

It is clear that (μc)c=μ. For a spectral measure E(·) on T, we have the family of measures {μx,yc:x,y∈H} defined via (3.2).

For the rest of this paper, we use M+(T) to denote the set of all finite positive Borel measures on T.

Proposition 3.3

Suppose μ∈M+(T) and μc≪μ. Then the following hold. μ≪μc;

The Radon–Nikodym derivatives satisfy dμcdμ(ξ)·dμcdμ(ξ¯)=1forμ-a.e.ξ∈T.

Proof

Let h=dμc/dμ. Observe that μ=(μc)c≪μc anddμ(ξ)=dμc(ξ¯)=h(ξ¯)dμ(ξ¯)=h(ξ¯)dμc(ξ)=h(ξ¯)h(ξ)dμ(ξ)forμ-a.e.ξ∈T.

Therefore,3.4 dμ(ξ)=h(ξ¯)dμc(ξ)andh(ξ¯)h(ξ)=1forμ-a.e.ξ∈T.

which completes the proof. □

The following proposition, originally explored in [16] for symmetric conjugations, relates a C∈Cc(U) with the associated spectral measure E(·) for U. Define Ec(·) on Borel subsets Ω of T by Ec(Ω):=E(Ω∗). From this definition it follows that ⟨Ec(Ω)x,y⟩=μx,yc(Ω) for all x,y∈H.

Proposition 3.5

Let C be a conjugation on H and U be a unitary operator on H with associated spectral measure E(·). Then we have the following. Ec(·) is the associated spectral measure for U∗.

CE(·)C is the spectral measure for CU∗C.

CUC=U∗ if and only if CE(Ω)C=E(Ω) for all Borel subsets Ω of T.

CUC=U if and only if CE(Ω)C=Ec(Ω) for all Borel subsets Ω of T.

Proof

If E(·) is a spectral measure, one can check that Ec(·) and CE(·)C are also spectral measures. Since, for each pair x,y∈H,⟨U∗x,y⟩=∫ξ¯d⟨E(ξ)x,y⟩=∫ξd⟨Ec(ξ)x,y⟩,

the uniqueness of the spectral measure for a unitary operator gives (a). In a similar way, (b) is a consequence of the computation⟨CU∗Cx,y⟩=⟨Cy,U∗Cx⟩=⟨UCy,Cx⟩=∫ξd⟨E(ξ)Cy,Cx⟩=∫ξd⟨x,CE(ξ)Cy⟩=∫ξd⟨CE(ξ)Cx,y⟩.

Note the use of (2.1) in the above calculation. To see (c), note that CU∗C=U if and only if their spectral measures CE(·)C and E(·) coincide. Symmetrically in (d), CU∗C equals to U∗ if and only if the spectral measures CE(·)C and Ec(·) coincide. □

As we will see in subsequent sections, the set Cc(U) is quite large and so an important step in understanding it is to decompose each C∈Cc(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 an A∈B(H) if AM⊆M; reducing if both AM⊆M and A∗M⊆M; and hyperinvariant if TM⊆M for every T∈B(H) that commutes with A. We begin with a simple lemma whose proof follows from (2.1) and the fact that C2=I.

Lemma 3.6

If C is a conjugation on H and M is a subspace of H such that CM⊆M, then CM=M and CM⊥=M⊥.

Proposition 3.7

Let U∈B(H) be a unitary operator with associated spectral measure E(·) and Ω⊆T be a Borel set. If Ω∗=Ω then for any C∈Cc(U), we have C(E(Ω)H)=E(Ω)H.

If Cc(U)≠∅ and E(Ω)H is invariant for C, then E(Ω\Ω∗)=0.

Proof

For the proof of (a), let x∈E(Ω)H and y∈(E(Ω)H)⊥. By Proposition 3.5(d) we have⟨Cx,y⟩=⟨CE(Ω)x,y⟩=⟨E(Ω∗)Cx,y⟩=⟨Cx,E(Ω)y⟩=⟨Cx,0⟩=0

and thus Cx∈E(Ω)H. Now apply Lemma 3.6.

For the proof of (b) let x∈E(Ω\Ω∗)H. From E(Ω)=E(Ω∗)⊕E(Ω\Ω∗), we can use Proposition 3.5(d) to see that0=‖E(Ω∗)x‖=‖CE(Ω)Cx‖=‖E(Ω)Cx‖=‖Cx‖=‖x‖.

□

For a unitary operator U on H, one can show, as was done in [19], that for any μ∈M+(T) the setHμ:={x∈H:μx≪μ}

is a reducing subspace of U. The space Hμ was explored in [17, §65] as part of a general discussion of the multiplicity theory for unitary operators.

Theorem 3.8

Let U be a unitary operator on H, E(·) its associated spectral measure, μ∈M+(T), and C∈Cc(U). Then we have the following. CHμ=Hμc and CHμ⊥=Hμc⊥, and thus

C=Cμ,μc⊕Cμ,μc′, where Cμ,μc=C|Hμ:Hμ→Hμc and Cμ,μc′=C|Hμ⊥:Hμ⊥→Hμc⊥ are antilinear, onto, isometries.

Proof

Let x∈Hμ. By Proposition 3.5(d), CE(·)C=Ec(·) and thus⟨E(·)Cx,Cx⟩=⟨x,CE(·)Cx⟩=⟨x,Ec(·)x⟩=⟨Ec(·)x,x⟩.

Since ⟨E(·)x,x⟩≪μ, it follows that ⟨E(·)Cx,Cx⟩≪μc and thus Cx∈Hμc. Similarly, CHμc⊆Hμ, thus CHμ=Hμc and CHμ⊥=Hμc⊥ (Lemma 3.6). □

Recall [17, §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}.

For a unitary U, there exists a scalar spectral measure ν, meaning that ν(Δ)=0 if and only if E(Δ)=0 [5, p. 293] (also see the discussion in Theorem 3.8 in [19]). For ν1,ν2∈M+(T) it was shown in [19, Prop. 3.10] that Hν1⊆Hν2 if and only if ν1∧μ≪ν2∧μ.

Corollary 3.9

Let U be a unitary operator on H and ν be any scalar spectral measure for U. Suppose that μ∈M+(T) satisfies μc∧ν≪μ∧ν. If C∈Cc(U), we have the following. CHμ=Hμ and CHμ⊥=Hμ⊥.

C=Cμ⊕Cμ⊥, where Cμ:=C|Hμ and Cμ⊥=C|Hμ⊥.

Cμ∈Cc(U|Hμ) and Cμ⊥∈Cc(U|Hμ⊥).

Corollary 3.10

Let U be a unitary on H and ν be any scalar spectral measure for U. Fix a μ∈M+(T). If CHμ⊆Hμ for some C∈Cc(U) then μc∧ν≪μ∧ν.

Proof

By Theorem 3.8 we have CHμ=Hμc⊆Hμ. Thus, by [19, Prop. 3.11], we obtain μc∧ν≪μ∧ν. □

Since a unitary operator is normal, we see that ker(U-αI)=ker(U∗-α¯I) i.e., Hδα=Hδαc, where δα denotes an atomic measure with atom at α∈T. This gives us the following corollary.

Corollary 3.11

Let U be a unitary operator on H and C∈Cc(U). Let α∈T be an eigenvalue for U. Then C=Cδα⊕Cδα⊥, where Cδα=C|Hδα and Cδα⊥=C|Hδα⊥ are conjugations on ker(U-αI) and ker(U-αI)⊥, respectively.

Natural conjugations on vector valued L2 spaces

This section provides a model for conjugations on vector valued Lebesgue spaces and will be useful in our description of Cc(U) in Theorem 5.3.

For a Hilbert space H with norm ‖·‖H and a μ∈M+(T), consider the set L0(μ,H) of H-valued μ-measurable functions f on T and the setL2(μ,H):={f∈L0(μ,H):‖f‖L2(μ.H):=(∫T‖f(ξ)‖H2dμ(ξ))12<∞}.

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) by(MUf)(ξ)=U(ξ)f(ξ)

for f∈L2(μ,H) and μ-a.e.  ξ∈T. Clearly MU∈B(L2(μ,H)). If we use the notation U∗(ξ)=U(ξ)∗, one can verify that4.1 MU∗=MU∗.

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

for f∈L2(μ,H) and μ-a.e.  ξ∈T. The case when φ(ξ)=ξ will play an prominent role in this paper in which case we have the vector-valued bilateral shift Mξ on L2(μ,H).

Recall from Sect. 2 that AB(H) denotes the space of all bounded antilinear operators on H. We define L∞(μ,AB(H)) to be the space of all μ-essentially bounded and AB(H)-valued Borel functions on T. Similarly as above, for C∈L∞(μ,AB(H)), define(ACf)(ξ)=C(ξ)f(ξ)

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

For any conjugation J on H, define the conjugation J on L2(μ,H) by(Jf)(ξ)=J(f(ξ)),f∈L2(μ,H).

Notice that JMξJ=Mξ¯ [19].

We now focus our attention on the scalar valued L2(μ) space and the set Cc(Mξ). This next result shows that when Cc(Mξ)≠∅, there must be some restrictions on μ. The set Cc(Mξ) was explored in [4] when μ=m.

Proposition 4.2

Let μ∈M+(T) and C be a conjugation on L2(μ) such that CMξ=MξC. Then μc≪μ (and hence μ≪μc by Proposition 3.3).

Proof

From (2.5), the identity CMξC=Mξ implies that CMξ¯C=Mξ¯. For any trigonometric polynomial p(ξ) define p#(ξ):=p(ξ¯)¯. The above (and the antilinearity of C) shows that CMpC=Mp#. Therefore, by the weak-∗ density of the trigonometric polynomials in L∞(μ), we obtain4.3 CMφC=Mφ#for anyφ∈L∞(μ),

where φ#(ξ)=φ(ξ¯)¯. If μc≪̸μ, then there is a Borel set Ω⊆T such that μ(Ω)≠0 but μc(Ω)=0. But, (4.3) yields a contradiction with φ=χΩ, since MχΩ∗=0 but CMχΩC is not. □

Now let us focus on the situation when μc≪μ. In this case we also have that μ≪μc (Proposition 3.3). For f∈L2(μ,H) and U∈L∞(μ,B(H)), it makes sense to write f(ξ¯) or U(ξ¯) and define4.4 U#(ξ):=U∗(ξ¯)=U(ξ¯)∗.

Proposition 4.5

Let μ∈M+(T) such that μc≪μ and let h=dμc/dμ. For a Hilbert space H, a conjugation J on H, and f∈L2(μ,H), define4.6 (J#f)(ξ)=(h(ξ))12J(f(ξ¯))

for μ-a.e.  ξ∈T. Then we have the following. J# is a conjugation on L2(μ,H);

J#MξJ#=Mξ.

Proof

As discussed in Proposition 3.3, μ≪μc and dμc=h#dμ with h#(ξ)h(ξ)=h(ξ¯)h(ξ)=1 for μ a.e.  ξ∈T.

Since J is antilinear on H, one sees that J# is antilinear on L2(μ,H). Moreover, for f∈L2(μ,H) we have‖J#f‖L2(μ,H)2=∫‖h(ξ)12J(f(ξ¯))‖H2dμ(ξ)=∫‖J(f(ξ¯))‖H2h(ξ)dμ(ξ)=∫‖f(ξ)‖H2h(ξ¯)dμ(ξ¯)=∫‖f(ξ)‖H2dμ(ξ)=‖f‖L2(μ,H)2.

Note the use of (3.4) above. Thus, J# is isometric on L2(μ,H).

Next we show that (J#)2=I. Indeed, for each f∈L2(μ,H),(J#J#f)(ξ)=h(ξ)12J((J#f)(ξ¯))=h(ξ)12J((h(ξ¯))12J(f(ξ)))=(h(ξ)h(ξ¯)12J(J(f(ξ)))=f(ξ).

Again, note the use of (3.4) above. Therefore, J# is a conjugation. To prove (b), observe that for each f∈L2(μ,H) we have(J#Mξf)(ξ)=J#(Mξf)(ξ)=h(ξ)12J((Mξf)(ξ¯))=h(ξ)12J(ξ¯f(ξ¯))=ξh(ξ)12J(f(ξ¯))=ξ(J#f)(ξ)=(MξJ#)f(ξ).

□

Remark 4.7

If μ=m, Lebesgue measure on T, then m=mc and h≡1 and the conjugation (4.6) coincides with the one considered in [4].

A special case worth pointing out is the scalar case H=C.

Corollary 4.8

Let μ∈M+(T) such that μc≪μ. Let h=dμc/dμ and define4.9 (J#f)(ξ)=h(ξ)12f(ξ¯)¯,f∈L2(μ).

Then J# is a conjugation on L2(μ) and J#MξJ#=Mξ.

In particular, observe that μc≪μ⟹Cc(Mξ)≠∅.

The following echos a result from [4, Proposition 4.2]. Recall the notation from (4.4).

Proposition 4.10

Let J be a conjugation on H, J# be defined by (4.6), and let U∈L∞(μ,B(H)) be a unitary operator-valued function. Then we have the following. J#MUJ#=JM(U#)∗J;

MUJ# is a conjugation on L2(μ,H) if and only if JU(ξ)J=U#(ξ)=U∗(ξ¯) for μ-a.e.  ξ∈T;

If MUJ# is a conjugation on L2(μ,H) then MUJ#=J#MU∗;

(MUJ#)Mξ(MUJ#)=Mξ.

Proof

For every f∈L2(μ,H), observe that for μ-a.e.  ξ∈T we have(J#MUJ#f)(ξ)=h(ξ)12J((MUJ#f)(ξ¯))=h(ξ)12J(U(ξ¯)(J#f)(ξ¯))=h(ξ)12J(U(ξ¯)h(ξ¯)12J(f(ξ)))=(h(ξ)h(ξ¯))12J(U(ξ¯)J(f(ξ)))=JU(ξ¯)J(f(ξ))=J((U#(ξ))∗J(f(ξ)))=(J(U#)∗Jf)(ξ).

Note the use of (4.4) above. This proves (a).

Note that MUJ# is antilinear and isometric on L2(μ,H). To prove that MUJ# is a conjugation (and thus complete the proof of (b)), Lemma 2.7 says that we just need to check the identity J#MUJ#=MU∗. By (a) this is equivalent to JU(ξ¯)J=U∗(ξ) since, by (4.1), (MU∗f)(ξ)=U∗(ξ)f(ξ).

Statement (c) follows from the fact that MUJ# is a conjugation on L2(μ,H), and so (MUJ#)(MUJ#)=I, along with the fact MUMU∗=MU∗MU=I (since U(ξ) is unitary for μ-a.e.  ξ∈T).

To see (d), observe that for any f∈L2(μ,H),(MUJ#Mξf)(ξ)=U(ξ)J#(Mξf)(ξ)=U(ξ)h(ξ)12J((Mξf)(ξ¯))=h(ξ)12U(ξ)J(ξ¯f(ξ¯))=ξh(ξ)12U(ξ)J(f(ξ¯))

while(MξMUJ#f)(ξ)=ξ(MUJ#f)(ξ)=ξU(ξ)(J#f)(ξ)=ξU(ξ)h(ξ)12J(f(ξ¯))=ξh(ξ)12U(ξ)J(f(ξ¯)),

which completes the proof of (d). □

Conjugations via the spectral theorem

In this section we use the multiplicity theory for unitary operators [5, 17] to describe Cc(U). We also prove that Cc(U)≠∅ if and only if U≅U∗ (thus establishing the converse to Lemma 2.9). We begin with a statement of the spectral multiplicity theory from [5, p.  307, Ch. IX, Theorem 10.20]. Recall the statement of the multiplicity version of the spectral theorem from (1.3).

Remark 5.1

Let U be a unitary operator with a spectral measure E(·). As previously observed in Proposition 3.5(a), Ec(·) is a spectral measure for U∗. In [19, Theorem 8.1], the measures μ∞,μ1,μ2,⋯ from (1.3) were constructed using the spectral measure E(·). Therefore, the appropriate measures for operator U∗ are μ∞c,μ1c,μ2c,⋯.

Lemma 5.2

Let U be a unitary operator on H with the multiplicity representation of U given by the mutually singular measures μ∞,μ1,μ2,⋯ as in (1.3). If Cc(U)≠∅, then μic≪μi for all i=∞,1,2,⋯.

Proof

Lemma 2.9 says that if Cc(U)≠∅, then U≅U∗. Hence, by Remark 5.1 and [5, p. 305, Theorem IX 10.16], the measures μi and μic are mutually absolutely continuous for all i=∞,1,2,⋯. □

We now arrive at the description of Cc(U) in terms of the parameters of the spectral theorem.

Theorem 5.3

Let U be a unitary operator and C be a conjugation on H. With the notation as in (1.3), assuming that μic≪μi for i=∞,1,2,⋯, the following are equivalent C∈Cc(U);

For each i=∞,1,2,⋯, there are conjugations Ci∈AB((L2(μi,Hi)) such that Mξ(i) is Ci–commuting and C=I∗(⨁Ci)I;

For each i=∞,1,2,⋯ and any conjugation J(i) on Hi, there is a unitary operator-valued function U(i)∈L∞(μi,B(Hi)) such that J(i)U(i)(ξ)J(i)=U(i)(ξ)#forμia.e.ξ∈TandC=I∗(⨁U(i)J#(i))I=I∗(⨁U(i))(⨁J#(i))I.

Proof

To show (a) ⟹ (c), let M~ξ:=IUI∗∈B(LH2) and define the conjugation C~=ICI∗ (note the use of Lemma 2.3). Then C~M~ξC~=M~ξ.

Let J(i) be any a conjugation on Hi. Since μic≪μi for i=∞,1,2,⋯, let hi=dμic/dμi and define the map J#(i) on L2(μi,Hi) by(J#(i)fi)(ξ)=hi(ξ)12J(i)(fi(ξ¯))

for μi-a.e.  ξ∈T and fi∈L2(μi,Hi). By Proposition 4.5, each of the above maps defines a conjugation on L2(μi,Hi) which satisfiesJ#(i)Mξ(i)J#(i)=Mξ(i).

Use these conjugations to define the conjugation J~#=⨁J#(i) on LH2 and observe thatJ~#M~ξJ~#=M~ξ.

Moreover,M~ξC~J~#=C~M~ξJ~#=C~J~#M~ξ.

The spectral theorem applied to Mξ~ also yields the commutant [5, p.  307, Theorem 10.20], namely there areU(i)∈L∞(μi,B(Hi)),i=∞,1,2,⋯,

such thatC~J~#=⨁M~U(i)=M~U(∞)⊕M~U(1)⊕M~U(2)⊕⋯.

Since C~J~# is unitary, it follows that M~U(i) is also unitary and consequently U(i) is a operator-valued function such that U(i)(ξ) is unitary for μi a.e.  ξ∈T. Therefore,C~=(⨁MU(i))(⨁J#(i))=⨁MU(i)J#(i).

Since C~|L2(μi,Hi) is a conjugation, it follows thatJ(i)U(i)(ξ)J(i)=(U(i)(ξ))#

for μi-a.e.  ξ∈T (Proposition 4.10). This completes the proof of (a) ⟹ (c).

To prove (c) ⟹ (b), it is enough to take C(i)=MU(i)J#(i). The remaining implication (b) ⟹ (a) is trivial. □

The following yields the converse of Lemma 2.9 and thus completes the criterion as to when Cc(U)≠∅.

Corollary 5.4

If U is a unitary operator on H such that U≅U∗, then there is conjugation C on H such that CUC=U.

Proof

If U≅U∗, then, as in the proof of Lemma 5.2, the measures μi and μic are mutually absolutely continuous for all i=∞,1,2,⋯. Now invoke Theorem  5.3 with any conjugation J(i) on Hj (and U(i)=IHi) and observe that the conjugation C=I∗J#I=I∗⨁J#(i)I commutes with U. □

Examples

In this section we use our results to give concrete descriptions of Cc(U) when U is a unitary matrix, multiplication by an inner function on L2(m), the Fourier transform, and the Hilbert transform.

Unitary matrices

For an n×n unitary matrix U, the condition as to when Cc(U) is nonempty, along with the description of Cc(U), can certainly be derived from Theorem 5.3. However, we give a simple proof using basic linear algebra. We begin with the following result from [15, Lemma 3.2].

Proposition 6.1

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 by6.2 J[x1x2⋯xn]t=[x1¯x2¯⋯xn¯]t,

i.e., C[x1x2⋯xn]t=V[x1¯x2¯⋯xn¯]t.

The spectral theorem for unitary matrices says that if U≅U∗, then U is unitarily equivalent to6.3 U′=ξ1In1ξ1¯In1⋱ξdIndξd¯IndIℓ-Ik,

where ξ1,…,ξd∈T\{1,-1} are distinct eigenvalues of U, Im denotes the m×m identity matrix, and the block in the lower right corner might not appear or might appear as just Iℓ or just -Ik, depending on whether 1 or -1 are eigenvalues of U. Of course nj, ℓ, and k represent the multiplicities of the respective eigenvalues and 2n1+⋯+2nd+ℓ+k=n. One can now use Proposition 6.1 to prove the following.

Theorem 6.4

Suppose that U is an n×n unitary matrix with U≅U∗ and W is a unitary matrix such that WUW∗=U′, where U′ is the matrix from (6.3). Then every C∈Cc(U) takes the formC=WV1V1t⋱VdVdtQℓQkJW∗,

where each Vj is an nj×nj unitary matrix, Qℓ, Qk are ℓ×ℓ and k×k (respectively) unitary matrices with Qℓt=Qℓ and Qkt=Qk (in which only one or perhaps both might not appear depending whether 1 or -1 are eigenvalues of U), and J is the conjugation on Cn from (6.2).

Unitary multiplication operators on L2(m,T)

As discussed in [20, Example 5.16] there is 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 dimension of any wandering subspace for U. In this section, we give a concrete description of Cc(Mψ). If J# is the conjugation on L2 defined by (J#f)(ξ)=f#(ξ)=f(ξ¯)¯, and C∈Cc(Mψ), then CJ# is a unitary operator on L2 for which (CJ#)Mψ=Mψ(CJ#). This trick was used in several places [3, 4, 7]. The bounded operators on L2 which commute with Mψ, i.e., the commutant of Mψ, were described in [19, Theorem 7.3].

Recall the known fact (see for example [20, Proposition 5.17]) that for an inner function ψ we have the following orthogonal decomposition for L2, namely,6.5 L2=⨁n=-∞∞ψnKψ,

where Kψ:=H2∩(ψH2)⊥ is the model space associated with ψ (see [10] for a review of model spaces).

Let us set up some notation to be used below. For an inner function ψ, let N:=dimKψ∈N∪{∞} and {hj}1⩽j⩽N be a fixed orthonormal basis for Kψ. Observe that N is finite if and only if ψ is a finite Blaschke product with N zeros, repeated according to multiplicity [10, Prop. 5.19]. Also define⨁1⩽j⩽NL2=L2⊕L2⊕⋯⊕L2.

The norm of an f=[fj]1⩽j⩽Nt of ⨁1⩽j⩽NL2 is ‖f‖:=(∑1⩽j⩽N‖fj‖22)12. When N=∞, we need to assume that the sum defining ‖f‖ above is finite. Furthermore, the operator ⨁1⩽j⩽NMξ (called the inflation of the bilateral shift Mξ on L2) is given by(⨁1⩽j⩽NMξ)f(ξ)=ξf(ξ)=[ξfj(ξ)]1⩽j⩽Nt.

We also defineℓN2:={x=[xj]1⩽j⩽Nt,xj∈C:‖x‖ℓN2=(∑1⩽j⩽N|xj|2)12<∞}.

When N=∞, this is the familiar sequence space ℓ2. Finally, observe that6.6 ⨁1⩽j⩽NMξ≅Mξ|L2(m,ℓN2).

As a consequence, using the discussion from Sect. 4, note that6.7 Cc(Mψ)≠∅.

We will actually describe Cc(Mψ) below.

From [19] we have the unitary operator6.8 W:L2→⨁1⩽j⩽NL2,Wf=[fj]1⩽j⩽Nt,

where f=∑1⩽j⩽Nhj·(fj∘ψ) is a unique decomposition given by [19, Lemma 7.3]. Note that6.9 fj=∑m=-∞∞amjξm

and the coefficients amj arise from the decomposition from (6.5) which yields the unique decomposition6.10 f=∑1⩽j⩽Nhj∑m=-∞∞amjψm.

Also recall from [19, Thm. 7.3] thatW∗[kj]1⩽j⩽Nt=∑1⩽j⩽Nhj·(kj∘ψ).

Let J and J# denote the standard conjugations on L2 defined by Jf(ξ)=f(ξ)¯ and (J#f)(ξ)=f#(ξ):=f(ξ¯)¯. For our inner function ψ, observe that ψ#=J#ψ is also inner.

Proposition 6.11

For an inner function ψ we have the following. J#Kψ=Kψ#.

If {hj}1⩽j⩽N is an orthonormal basis for Kψ then {hj#}1⩽j⩽N is an orthonormal basis for Kψ#.

Proof

Part (a) was shown in [3, Lemma 4.4] while part (b) is a consequence of the facts that conjugations preserve orthonormality (recall (2.1)). □

Let W# be the unitary operator from (6.8), where the inner function ψ is replaced by ψ# and orthonormal basis and the orthonormal basis {hj}1⩽j⩽N is replaced by the orthonormal basis {hj#}1⩽j⩽N, i.e.,W#g=[gj]1⩽j⩽Nt,whereg=∑1⩽j⩽Nhj#·(gj∘ψ#).

There are the two natural conjugations J and J# on ⨁1⩽j⩽NL2 defined for each F∈⨁1⩽j⩽NL2, F=[fj]1⩽j⩽Nt, byJF=[f¯j]1⩽j⩽Nt=:F¯andJ#F=[fj#]1⩽j⩽Nt=:F#.

Proposition 6.12

Let ψ be an inner function and {hj}1⩽j⩽N be an orthonormal basis for Kψ. Then we have the following. If f=∑1⩽j⩽Nhj·(fj∘ψ) then f#=J#f=∑1⩽j⩽Nhj#·(fj#∘ψ#).

W#J#W∗=J#.

Proof

Let f∈L2 and observe from (6.9) and (6.10) thatJ#f=∑j=1∞hj#∑amj¯(ψ#)mandfj#=∑m=-∞∞amj¯ξm.

Hence J#f=∑j=1∞hj#·(fj#∘ψ#), which proves (a). The above also yieldsW#J#W∗[fj]1⩽j⩽Nt=W#J#f=[fj#]1⩽j⩽Nt=J#[fj]1⩽j⩽Nt,

which proves (b). □

Theorem 6.13

Suppose that ψ is inner and {hj}1⩽j⩽N is an orthonormal basis for Kψ. Then we have the following, Cc(Mψ)≠∅.

C∈Cc(Mψ) if and only if there is a Φ=[φij]1⩽i,j⩽N∈L∞(m,ℓN2) such that 6.14 Φ(ξ)=Φ(ξ¯)andΦ∗(ξ)Φ(ξ)=Ia.e.onTand

6.15 Cf=∑1⩽j⩽N(fj#∘ψ)∑1⩽k⩽Nhk·(φk,j∘ψ)

for all f=∑1⩽j⩽Nhj·(fj∘ψ)∈L2.

Proof

Statement (a) follows from (6.7). To prove (b), observe that since W is a unitary operator, then C~:=WCW∗ is a conjugation on ⨁1⩽j⩽NL2 (Lemma 2.3). If CMψ=MψC, it follows from [19, Theorem 7.2(c)] thatC~(⨁1⩽j⩽NMξ)=(⨁1⩽j⩽NMξ)C~.

Since the operator ⨁1⩽j⩽NMξ on ⨁j⩾1L2 is unitary equivalent to Mξ on L2(m,ℓN2) (recall (6.6)), the result [4, Theorem 4.3] says there is a Φ=[φij]1⩽i,j⩽N∈L∞(m,ℓN2) such that Φ(ξ) is unitary for a.e. ξ∈T, MΦ is J#–symmetric, and C~=MΦJ#. The unitary property gives Φ∗(ξ)Φ(ξ)=I and the J#–symmetry property gives Φ(ξ)=Φ(ξ¯) a.e.  on T. So far, we have shown that if C is a conjugation which commutes with Mψ, then WCW∗=MΦJ#, where Φ satisfies the two properties from (6.14). Conversely suppose that Φ=[φij]1⩽i,j⩽N∈L∞(m,ℓN2) satisfies the two conditions from (6.14). The second condition will show that J#MΦJ#=MΦ∗ and combining this with the first condition will show that Φ is unitary valued a.e. The second property, along with Proposition 4.10 will show that MΦJ# is a conjugation and belongs to Cc(Mξ). By the discussion above, this says that W∗(MΦJ#)W∈Cc(Mψ).

Applying Proposition 6.12 we can verify the formula (6.15). Indeed, for each f=∑1⩽j⩽Nhj·(fj∘ψ)∈L2 we haveCf=W∗MΦJ#Wf=W∗MΦW#J#(∑1⩽j⩽Nhj·(fj∘ψ))=W∗MΦW#(∑1⩽j⩽Nhj#·(fj#∘ψ#))=W∗[φij]1⩽i,j⩽N[fj#]1⩽j⩽Nt=W∗[∑1⩽i,j⩽Nφ1jfj#,∑1⩽i,j⩽Nφ2jfj#,∑1⩽i,j⩽Nφ3jfj#,…]t=h1·(∑1⩽i,j⩽Nφ1jfj#)∘ψ+h2·(∑1⩽i,j⩽Nφ2jfj#)∘ψ+…=(f1#∘ψ)·(∑1⩽i,j⩽Nhj(φj1∘ψ)+(f2#∘ψ)·(∑1⩽i,j⩽Nhj(φj2∘ψ)+⋯

and this completes the proof. □

Example 6.16

Consider the inner function ψ(z)=z. Here the associated unitary operator Mψ is merely the bilateral shift Mξ on L2. In this case, Kψ=C (the constant functions). Moreover, ψ#(z)=z and the expansions from Proposition 6.12 are the standard Fourier expansions of an f∈L2. Theorem 6.13 says that any C∈Cc(Mξ) takes the form (Cf)(ξ)=u(ξ)f(ξ¯)¯ for some u∈L∞ that is unimodular and satisfies u(ξ)=u(ξ¯) a.e.  on T.

Example 6.17

Consider the inner function ψ(z)=z2 as in [19, Example 7.7]. Then Kψ=span{1,z}={h1,h2}. Furthermore, using the notation from this section,f(ξ)=h1(ξ)f1(ξ2)+h2(ξ)f2(ξ2)=f1(ξ2)+ξf2(ξ2),

wheref1(ξ)=∑j=-∞∞f^(2j)ξjandf2(ξ)=∑j=-∞∞f^(2j+1)ξj.

From here, one can check (Theorem 6.13) that every C∈Cs(Mξ2) takes the form(Cf)(ξ)=f1#(ξ2)(φ11(ξ2)+ξφ21(ξ2))+f2#(ξ2)(φ12(ξ2)+ξφ22(ξ2)),

where φij are bounded measurable functions on T for which6.18 φ11(ξ)¯φ21(ξ)¯φ12(ξ)¯φ22(ξ)¯φ11(ξ)φ12(ξ)φ21(ξ)φ22(ξ)=1001.

and φij(ξ¯)=φij(ξ), i,j=1,2, for a.e.  ξ∈T. Condition (6.18) is equivalent to the conditions|φ11(ξ)|2+|φ21(ξ)|2=1,|φ12(ξ)|2+|φ22(ξ)|2=1,φ11(ξ)¯φ12(ξ)+φ21(ξ)¯φ22(ξ)=0.

Fix the convention that t=Arg(ξ)∈(-π,π] and that s(t),α(t),β(t),γ(t),δ(t) are any 2π–periodic real-valued bounded measurable functions. Considering the moduli of the functions above, we obtain0⩽s(t)⩽1,φ11(ξ)=eiα(t)s(t),φ12=eiβ(t)1-s2(t),φ21(ξ)=eiγ(t)1-s2(t),φ22(ξ)=eiδ(t)s(t).

As to the arguments of the functions above, we obtainδ(t)=β(t))+γ(t)-α(t)-π.

Incorporating the conditions φij(ξ¯)=φij(ξ), i,j=1,2, we obtainφ11(ξ)φ12(ξ)φ21(ξ)φ22(ξ)=eiα(|t|)s(|t|)eiβ(|t|)1-s2(|t|)eiγ(|t|)1-s2(|t|)-ei(β(|t|)+γ(|t|)-α(|t|))s(|t|).

Finally, every conjugation C∈Cs(Mξ2) must take the form(Cf)(ξ)=f1#(ξ2)(eiα(2|t|)s(2|t|)+ξei(γ(2|t|))1-s2(2|t|))+f2#(ξ2)(eiβ(2|t|)1-s2(2|t|)-ξei(β(2|t|)+γ(2|t|)-α(2|t|))s(2|t|)),

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

Example 6.19

As a specific nontrivial example of a C∈Cc(Mξ2) we can take(Cf)(ξ)=f1#(ξ2)(sin(2|t|)+ξcos(2t))+f2#(ξ2)(cos(2t)-ξsin(2|t|)),

where t=Arg(ξ) and s(t)=sin(t),α(t)≡0,β(t)≡0,γ(t)≡0. For another nontrivial example of a C∈Cc(Mξ2), we set s(t)≡s∈[0,1],α(t)=λt,λ∈R,β(t)≡0,γ≡0 to get(Cf)(ξ)=f1#(ξ2)(seiλ|t|+ξ1-s2)+f2#(ξ2)(1-s2-ξeiλ|t|)),

where t=Arg(ξ).

The bilateral shift on a L2(μ,T) space

In Example 6.16Cc(Mξ) for the bilateral shift Mξ on L2(m,T). This example contains a description of Cc(U) when U=Mξ on a more complicated L2(μ,T) space. Let g:[-1,1]→[0,∞) be defined piecewise byg(t)=32t2,t∈[0,1],52t4,t∈[-1,0].

If dt represents Lebesgue measure on [-1,1], define the following measures on the Borel subsets Ω⊆[-1,1] byμ~1(Ω)=∫Ωg(t)dt,μ~2(Ω)=∫Ωg(-t)dt.

One can verify thath~(t):=dμ~2dμ~1(t)=dμ~2dt(dμ~1dt)-1(t)=53t2,t∈[0,1],35t-2,t∈[-1,0].

Clearly h~(t)·h~(-t)=1 on [-1,1]. Now letγ:[-1,1]→T,γ(t)=exp(2πit)

and check that γ-1(ξ)=Argξ2π for ξ∈T. Define measures μ1,μ2∈M+(T) on Borel sets Ω⊆T byμk(Ω)=μ~k(γ-1(Ω)),k=1,2,

and observe that μ1c=μ2 and μ2≪μ1. Moreover, we can write the Radon-Nikodym derivativeh(ξ):=dμ2dμ1(ξ)=h~(γ-1(ξ))=(53)sgn(Argξ)(Argξ)2sgn(Argξ).

From here one sees that h(ξ)h(ξ¯)=1 on T as seen in Proposition 3.3.

As in (4.9), define J# on L2(μ1) by (J#f)(ξ)=h(ξ)12f(ξ¯)¯. Then J# is a conjugation and J#MξJ#=Mξ. Moreover, Theorem 5.3 says that any conjugation C on L2(μ1) such that CMξC=Mξ can be written as C=uJ#, where u∈L∞(μ1) is unimodular and u(ξ¯)=u(ξ) for μ1 a.e.  ξ∈T.

The Fourier transform

Let F denote the standard Fourier–Plancherel transform on L2(R). It is well known that F is unitary with spectrum {1,i,-1,-i}. Moreover, the Hermite functions {Hn}n⩾0 form an orthonormal basis for L2(R) and FHn=(-i)nHn for all n⩾0, i.e., the Hermite functions form an eigenbasis for F [14, Ch.11]. A description of Cs(F), the symmetric conjugations, was given in [19, Example 4.3]. In this example we work out Cc(F), the commuting conjugations for F. We first note that F≅F∗ (Example 2.4). Thus, Cc(F)≠∅ (Corollary 5.4).

To describe Cc(F), we proceed as follows. Our discussion so far says that6.20 L2(R)=E1⊕E-i⊕E-1⊕Ei,

where Eα=ker(F-αI). Define a conjugation J on L2(R) for which JHn=Hn for all n⩾0 (initially define J on Hn by JHn=Hn and extend antilinearly to all of L2(R)).

If ℓ2=ℓ2(N0) is the classical sequence space with the standard orthonormal basis {en}n⩾0, andV=V1⊕V-i⊕V-1⊕Vi,

where V1 is the unitary from E1 to ℓ2 defined by V1(H4n)=en; V-i is the unitary from E-i to ℓ2 defined by V-i(H4n+1)=en; V-1 is the unitary from E-1 to ℓ2 defined by V-1(H4n+2)=en; Vi is the unitary from Ei to ℓ2 defined by Vi(H4n+3)=en; then V is a unitary operator from L2(R) onto L2(μ,ℓ2), where μ=δ1+δ-i+δ-1+δi.

Define a conjugation J~ on ℓ2 by J~(en)=en for all n⩾0. Since μc≪μ we can define a conjugation J~# on L2(μ,ℓ2) such that (VFV∗)J~#=J~#(VFV∗) by (4.6). In other words, with respect to the orthogonal decompositionL2(μ,ℓ2)=L2(δ1,ℓ2)⨁L2(δ-i,ℓ2)⨁L2(δ-1,ℓ2)⨁L2(δi,ℓ2),

the conjugation J~# can be written in matrix form asJ~#=J~000000J~00J~00J~00.

The conjugation V∗J#V commutes with F and it can be written with respect to the Hermite basis asJ#H4n+k:=W∗J~#WH4n+k=H4n+k,k=0,H4n+k+2,k=1,H4n+k,k=2,H4n+k-2,k=3.

Therefore, the matrix representation of J# with respect to the orthogonal decomposition in (6.20) isJ#=J000000J00J00J00.

Moreover, by Theorem 5.3, any conjugation C~ on L2(μ,ℓ2) such thatC~(VFV∗)=(VFV∗)C~

can be represented by the matrixC~=U~1J~000000U~-iJ~00U~-1J~00U~iJ~00,

where U~1,U~-i,U~-1,U~i, are unitary operators on ℓ2 andJ~U~1J~=U~1∗,J~U~-1J~=U~-1∗,J~U~iJ~=U~-i∗.

The first two identities say that the unitary operators U~1 and U~-1 are represented by with respect to the basis {en}n⩾0 by a matrix with real entries. The last identity says that the matrix representations in the basis {en}n⩾0 of U~i and U~-i satisfy ⟨U~-iem,en⟩=⟨U~i∗em,en⟩,¯ which we write as U~-i=U~i#. Therefore, any conjugation C~ on L2(μ,ℓ2) such that C~(VFV∗)=(VFV∗)C~ can be represented asC~=U~1RJ~000000U~i#J~00U~-1RJ~00U~iJ~00,

where U1R and U-1R are arbitrary unitary operators on ℓ2 whose matrix representations with respect to {en}n⩾0 have real entries and U~i is arbitrary. Finally, a conjugation C on L2(R) fulfils the condition CF=FC if and only if it is represented with respect to the decomposition in (6.20) asC=U1RJ000000Ui#J00U-1RJ00UiJ00=U1R000000Ui#00U-1R00Ui00J,

where U1R,U-1R are arbitrary unitary operators on their respective eigenspaces ker(F-I) and ker(F+I) which are represented in terms of the basis {H4n}n⩾0 and {H4n+2}n⩾0 by real matrices, Ui is an arbitrary unitary operator on ker(F+iI) and Ui# is the unitary operator on ker(F-iI) defined by ⟨Ui#H4m+3,H4n+3⟩=⟨Ui∗H4m+1,H4n+1⟩¯,m,n⩾0.

The Hilbert transform

Suppose H is the Hilbert transform on L2(R). Since the spectrum of H is {i,-i} then L2(R)=Ei⨁E-i and Ei has orthonormal basis Bi={fn}n⩾1fn(x)=1π(x+i)n-1(x-i)n

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

See [14, Ch. 12] for details. In this example we will describe Cc(H). Note first that Cc(H)≠∅ (recall Example 2.4 and thus H≅H∗).

Similarly as was done with the Fourier transform, we can identify L2(R) with L2(μ,ℓ2), where μ=δi+δ-i. Then, the conjugation J# given by equality (4.6) is an antilinear extension of operator J#fn=gn,J#gn=fn,n⩾1. Putting this in matrix formJ#=0JJ0,

where J is a conjugation on L2(R) which fixes all elements of Bi and B-i.

Moreover, by Theorem 5.3, any conjugation C on L2(R) with CHC=H∗ must take the (block) formC=Ui00Ui#0JJ0=0UiJUi#J0,

where Ui is arbitrary unitary operator on Ei and Ui#∈B(E-i) defined as ⟨Ui#gm,gn⟩=⟨Ui∗fm,fn⟩¯ similar to the previous example.

A remark about invariant subspaces

The first paper in this series [19] classified, for a fixed unitary operator U on H, the subspaces M of H for which CM⊆M for every Cs(U) (the symmetric conjugations for U). These turned out to be the hyperinvariant subspaces for U. What are the subspaces M for which CM⊆M for every C∈Cc(U) (the commuting conjugations for U)? We have seen some partial results in this paper (see for example Proposition 3.7 and Corollary 3.9). We present some positive result in this direction. However, we do not have so pleasant characterization as in symmetric case. Generally these subspaces seem complicated to be simply described in the abstract situation. The difficulties, which can be came across even for multiplication operator, will be seen in Example 7.4. Of course, we need to have the natural assumption that Cs(U)≠∅. We start a characterization in the special case where U=Mξ on L2(μ,H). Recall the notation from Proposition 4.5. Then we have to assume that μc≪μ.

Theorem 7.1

Suppose μ∈M+(T) such that μc≪μ and H is a Hilbert space. For a subspace K of L2(μ,H) the following are equivalent. CK⊆K for every C∈Cc(Mξ);

For any fixed conjugation J on H and J# defined as in (4.6), subspace K is invariant for J# and every MF, where F belongs to Lc∞(μ,B(H)):={F∈L∞(μ,B(H)):JF(ξ)J=F(ξ)#forμ-a.e.ξ∈T}.

The proof of Theorem 7.1 requires a decomposition theorem from [22, pf.  of Corollary 3.19]. We include a proof for completeness and since the particular form of the decomposition is important for the proof of Theorem 7.1.

Lemma 7.2

Any A∈B(H) can be expressed as a positive constant times the sum of four unitary operators on H.

Proof

DefineH=12‖A‖(A+A∗)andK=12i‖A‖(A-A∗)

and notice that H and K are selfadjoint contractions and thus I-H and I-K are positive and hence have unique positive square roots. Thus,U1,2=H±i(I-H2)12andU3,4=iK±(I-K2)12

are four unitary operators which satisfyA=‖A‖2(U1+U2+U3+U4).

□

Proof of Theorem 7.1

The proof of (b) ⟹ (a) follows from a special case of Theorem 5.3. For the proof of (a) ⟹ (b), we begin with the fact that since J#∈Cc(Mξ) then K is invariant for J#. Moreover, by Theorem 5.3 any C∈Cc(Mξ) can be written as C=MUJ# for some U∈L∞(μ,B(H)) that is unitary valued μ-a.e.  and satisfies JU(ξ)J=U(ξ)# for μ-a.e.  ξ. Thus, K is invariant for MUJ# and thus MU. Apply Lemma  7.2 to any F∈Lc∞(μ,B(H)) to see that K is invariant for every MF where F∈Lc∞(μ,B(H)). □

The example bellow shows that more pleasant characterization of subspaces which are invariant for all commuting conjugation, even in scalar case of multiplication operators, will be difficult to find.

Remark 7.3

In the scalar case H=C, note that L∞(μ,B(C))={v∈L∞(μ):v(ξ)=v(ξ¯)forμ-a.e.ξ∈T}.

Example 7.4

For the bilateral shift Mξ on L2=L2(m,T), we know that every C∈Cc(Mξ) takes the form MuJ, where (Jf)(ξ)=f(ξ) and u∈L∞ such that u(ξ) is unimodular and u(ξ)=u(ξ¯) a.e. One can check that examples of subspaces that are invariant for every C∈Cc(Mξ) include {g∈L2:g(eit)=0,|t|⩾π2,g(eit)=g(e-it),|t|<π2};

{g∈L2:g(eit)=0,|t|⩾π2,g(eit)=-g(e-it),|t|<π2};

{g∈L2:g(eit)=0,|t|⩾π2,g(eit)=g(e-it),π4<|t|<π2,g(eit)=-g(e-it),|t|<π4}.

The variety of these spaces convinces us that a concise description of the C-invariant subspaces for every C∈Cc(Mξ) seems difficult.

We finish with a characterization using the spectral multiplicities and Theorems 5.3 and 7.1

Theorem 7.5

Suppose U is a unitary operator on H with Cc(U)≠∅ and let K be a closed subspace of H. With the notation as in (1.3), the following are equivalent. CK⊆K for every C∈Cc(U);

IK=⨁Ki, Ki⊆Hi, where CiKi⊆Ki, for all i and all Ci∈Cc(Mξ(i));

IK=⨁Ki, Ki⊆Hi, and all i and for any fixed conjugation Ji on Hi and J#(i) defined as in (4.6), each subspace Ki is invariant for J#(i) and every MFi, where Fi belongs to Lc∞(μi,B(Hi)):={Fi∈L∞(μi,B(Hi)):JiFi(·)Ji=Fi(·)#,μia.e.}.

Proof

Lemma 5.2 says that μic≪μi for each i. Note also that K is invariant for C∈Cc(U) if and only if IK is invariant for ICI∗∈Cc(M~ξ). Moreover, by Theorem 5.3, each C∈Cc(U) can be expressed as C=I∗(⨁Ci)I, where Ci∈Cc(Mξ(i)). Observe that if ⨁Ci∈Cc(M~ξ), then ⨁ϵiCi∈Cc(M~ξ) for any choice ϵi=±1 and also recall that if subspace is invariant for a conjugation its orthogonal complement is also invariant. Therefore, IK=⨁Ki where CiKi⊆Ki for all Ci∈Cc(Mξ(i)). This proves the equivalence of (a) and (b). The equivalence of (b) and (c) follows from Theorem 7.1. □

Data availability

No datasets were generated or analysed during the current study.

Declarations

Conflict of interet

The authors declare no competing interests.

This work was supported by the NSERC Discovery Grant (Canada) and by the Ministry of Science and Higher Education of the Republic of Poland.

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References

1. Bender CM Making sense of non-hermitian hamiltonians Rep. Prog. Phys. 2007 70 6 947 1018 10.1088/0034-4885/70/6/R03
2. Bender CM Boettcher S Real spectra in non-hermitian hamiltonians having PT symmetry Phys. Rev. Lett. 1998 80 5243 5246 10.1103/PhysRevLett.80.5243
3. Cristina Câmara M Kliś-Garlicka K Ł anucha B Ptak M Conjugations in L2 and their invariants Anal. Math. Phys. 2020 10 2 14
4. Cristina Câmara M Kliś-Garlicka K Ł anucha B Ptak M Conjugations in L2(H) Integral Equ. Operator Theory 2020 92 6 25
5. Conway JB A course in functional analysis. Graduate Texts in Mathematics 1985 New York Springer-Verlag
6. Dymek P Płaneta A Ptak M Conjugations on L2(TN) and invariant subspaces Complex Anal. Oper. Theory 2022 16 7 12 10.1007/s11785-022-01251-6
7. Dymek P Płaneta A Ptak M Conjugations preserving Toeplitz kernels Integral Equ. Operator Theory 2022 94 4 18 10.1007/s00020-022-02714-3
8. Garcia S Conjugation, the backward shift, and Toeplitz kernels J. Operator Theory 2005 54 2 239 250
9. Garcia, S. R.: Conjugation and Clark operators. In: Recent Advances in Operator-Related Function Theory, vol. 393 of Contemp. Math., pages 67–111. Amer. Math. Soc., Providence, RI, (2006)
10. Garcia, S.R., Mashreghi, J., Ross, W.T.: Introduction to model spaces and their operators. Cambridge Studies in Advanced Mathematics, vol. 148. Cambridge University Press, Cambridge (2016)
11. Garcia SR Prodan E Putinar M Mathematical and physical aspects of complex symmetric operators J. Phys. A 2014 47 35 54 10.1088/1751-8113/47/35/353001
12. Garcia SR Putinar M Complex symmetric operators and applications Trans. Am. Math. Soc. 2006 358 3 1285 1315 10.1090/S0002-9947-05-03742-6
13. Garcia SR Putinar M Complex symmetric operators and applications. II Trans. Am. Math. Soc. 2007 359 8 3913 3931 10.1090/S0002-9947-07-04213-4
14. Garcia, S. R., Mashreghi, J., Ross, W. T.: Operator theory by example, vol. 30 , Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, (2023)
15. Garcia SR Tener JE Unitary equivalence of a matrix to its transpose J. Operator Theory 2012 68 1 179 203
16. Godich VI Lutsenko IE On the representation of a unitary operator in the form of a product of two involutions Uspehi Mat. Nauk 1965 20 64 65
17. Halmos Paul R Introduction to Hilbert Space and the Theory of Spectral Multiplicity 1951 New York Chelsea Publishing Co.
18. Ilišević D Ptak M Conjugations on Banach ∗-algebras Ann. Funct. Anal. 2020 11 4 1274 1286 10.1007/s43034-020-00085-7
19. Mashreghi, J., Ptak, M., Ross, W. T.: Conjugations of unitary operators, I. preprint
20. Mashreghi, J., Ptak, M., Ross, W. T.: A decomposition theorem for unitary operators. preprint
21. Ptak M Simik K Wicher A C-normal operators Electron. J. Linear Algebra 2020 36 67 79 10.13001/ela.2020.5045
22. Radjavi H Rosenthal P Invariant Subspaces 2003 2 Mineola, NY Dover Publications Inc
