Conjugations of unitary operators, II

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INTRODUCTION
This is the second in a series of two papers that explore conjugations of unitary operators on separable complex Hilbert spaces.The first paper [21] in this series 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 CU C = U * .An argument with the spectral theorem says there will always be a conjugation C with this property.Moreover, [21] contains several characterizations of the set of all such conjugations C for which CU C = U * .These conjugations are known as the "symmetric conjugations" for U .
The purpose of this second paper is to explore, for a given unitary operator U on H, the set (1.1) C c (U ) := {C is a conjugation on H : CU C = U }.
These are known as the "commuting conjugations" for U .The subscript c in the definition of C c (U ) might initially seem superfluous but we will use it anyway to distinguish this set from C s (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 (U f )(ξ) = ξf (ξ), the bilateral shift on L 2 (m, T), where m is normalized Lebesgue measure on the unit circle T. One can check that the map on L 2 (m, T) defines a conjugation which satisfies JU J = U .Moreover (see Example 7.12), any conjugation C on L 2 (m, T) for which CU C = U takes the form (Cf )(ξ) = u(ξ)(Jf )(ξ), where u ∈ L ∞ (m, T) is both unimodular and satisfies u(ξ) = u(ξ) almost everywhere on T.An analogous result holds when (U f)(ξ) = ξf(ξ) on the vector-valued Lebesgue space L 2 (m, H) (Theorem 6.4) but not always on L 2 (µ, H) for a general positive measure on T (see §5 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 CU C = U .Indeed, using the known fact from [17] (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 C c (U ) = ∅, then U ∼ = U * (i.e., U is unitarily equivalent to its adjoint U * ).One of the main results of this paper (Corollary 8.5) is the converse.
Theorem 1.2.For a unitary operator U on a complex separable Hilbert space H, the following are equivalent.
(a) C c (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, i.e., λ ∈ σ(U ) if and only if λ ∈ σ(U ) (since if U ∼ = U * then σ(U ) = σ(U * ) = σ(U )).Thus, as an example, for the bilateral shift (U f )(ξ) = ξf (ξ) on L 2 (µ, T), where µ is a finite positive Borel measure on T, a standard argument shows that σ(U ) = {ξ ∈ T : µ(I δ (ξ)) > 0 for all δ > 0} (I δ (ξ) is the arc of the circle centered at ξ of radius δ).Thus, for example, if the measure µ is supported on the top half of T, then there are no conjugations C on L 2 (µ, T) for which CU C = U .Of course, one could also consider the easy example of a unitary matrix whose eigenvalues are not symmetric with respect to the real axis.
For a unitary operator U on a finite dimensional Hilbert space, where we can regard, via a matrix representation with respect to an orthonormal basis, U as a unitary matrix, we can use the linear algebra version of the spectral theorem to see that U ∼ = U * if and only if where W is a unitary matrix, ξ 1 , . . ., ξ d ∈ T\{1, −1} are distinct eigenvalues of U , I m detnoes the m × m identity matrix, and the block in the lower right corner might not appear, or might appear as just I ℓ or just −I k , depending on whether 1 or −1 are eigenvalues of U .Of course n j , ℓ, and k represent the multiplicities of their respective eigenvalues.As we will prove in Theorem 3.6, such unitary matrices satisfy C c (U ) = ∅ and every C ∈ C c (U ) takes the form where each V j is an n j × n j unitary matrix, Q ℓ , Q k are ℓ × ℓ and k × k (respectively) unitary matrices with which only one or perhaps both might not appear depending whether 1 or −1 are eigenvalues of U ), and J is the conjugation on C n defined by Jx = x (complex conjugating each of the entries of x).

Another basic type of unitary operator is
where µ is a positive finite Borel measure on T and H is a Hilbert space (see §5 for the precise definitions).As discussed earlier, there might not be any commuting conjugations for U .In §5 we discuss the restrictions one must place on µ so that C c (U ) = ∅ and, when these conditions are satisfied, describe C c (U ).Since the operators f(ξ) → ξf(ξ) on these L 2 (µ, H) spaces are the building blocks for any unitary operator on a general Hilbert space, via the spectral theorem, we describe the commuting conjugations (when they exist) for a general unitary operator in §8 (where we also prove Theorem 1.2).
A particularly interesting class of unitary operators are the multiplication operators M ψ f = ψf on L 2 (m, T) where ψ is an inner function.Here one has the added connection to the theory of model spaces H 2 ∩ (ψH 2 ) ⊥ [11].As discussed in [22], these multiplication operators serve as models for general bilateral shifts.In §7 we show that M ψ is unitarily equivalent to its adjoint (and hence C c (M ψ ) = ∅ via Theorem 1.2) and proceed to give a concrete description of C c (M ψ ) (Theorem 7.9).
In the last section of this paper, we work out a concrete description of C c (F) for the classical Fourier-Plancherel transform F on L 2 (R) (Example 9.2) and a description of C c (H ) for the classical Hilbert transform H on L 2 (R) (Example 9.4).

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 + αCy 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 C 2 = I (C is involutive).By the polarization identity, a conjugation also satisfies (2.1) Cx, Cy = y, x for all x, y ∈ H.
Example 2.2.Many types of conjugations were outlined in [12,13,14].Below are a few basic ones that are relevant to this paper.
(a) The mapping Cf = f defines a conjugation on a standard Lebesgue space L 2 (µ, X).In particular, the mapping defines a conjugation on C n .Throughout this paper we will use the symbol t to represent the transpose of a matrix.In addition, vectors in C n will be viewed as column vectors since, for an n × n matrix A of complex numbers, we will often consider linear transformations on C n defined by x → Ax.
(c) On L 2 (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.Recalling the definition of C c (U ) from (1.1), let us make a few elementary observations.One can argue from (2.1) that Next we comment that the commuting conjugations are stable under unitary equivalence.
Proposition 2.6.Suppose U, V, W are unitary operators on If U is unitary and C is a conjugation on H, then U C ∈ AB(H) and is isometric.This next result has a straightforward proof and determines when U C is involutive and hence a conjugation.
Lemma 2.7.Let U be a unitary operator and C be a conjugation on H. Then U C is a conjugation if and only if CU C = U * .
We recall the following result from [17] (also see the proof of Proposition 2.5 from [21]) 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 J 1 and J 2 on H such that U = J 1 J 2 .Moreover, J 1 U J 1 = U * and J 2 U J 2 = U * .
In the introduction we showed that although every unitary operator U satisfies CU C = U * with respect to some conjugation C, it is possible for C c (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 8.5).
Lemma 2.9.If U is a unitary operator on H and C c (U ) = ∅, then U ∼ = U * .
Proof.Let J 1 be as in Proposition 2.8, C ∈ C c (U ), and define V = J 1 C. Clearly V is unitary (since it is linear, isometric, and onto) and

COMMUTING CONJUGATIONS OF UNITARY MATRICES
For an n × n unitary matrix U , the condition as to when C c (U ) is nonempty, along with the description of C c (U ), is straightforward and so we work it out in this separate section.We begin with the following result from [16,Lemma 3.2].
where V is an n × n unitary matrix with V t = V and J is the conjugation on C n defined by We now establish when C c (U ) = ∅ for an n × n unitary matrix U .This is a special case of Theorem 1.2.Proof.The implication (a) =⇒ (b) is from Lemma 2.9.For the implication (b) =⇒ (a), suppose that U ∼ = U * .As mentioned in the introduction, the spectral theorem for unitary matrices implies that U is unitarily equivalent to , where ξ 1 , . . ., ξ d ∈ T \ {1, −1} are distinct eigenvalues of U , I m denotes the m × m identity matrix, and the block in the lower right corner might not appear or might appear as just I ℓ or just −I k , depending on whether 1 or −1 are eigenvalues of U .Of course n j , ℓ, and k represent the multiplicities of the respective eigenvalues and Now consider the mapping where J is the conjugation on C n from (3.2).Proposition 3.1 says that C ′ is a conjugation on C n and block multiplication will show that Theorem 3.6.Suppose that U is an n × n unitary matrix with U ∼ = U * and W is a unitary matrix such that W U W * = U ′ , where U ′ is the matrix from (3.4).Then every C ∈ C c (U ) takes the form where each V j is an n j ×n j unitary matrix, Q ℓ , Q k are ℓ×ℓ and k×k (respectively) unitary matrices with , where U ′ is the matrix from (3.4).From Proposition 3.1, C ′ = V J, where V is an n × n unitary matrix with V t = V .Now observe that JU ′ J = U ′ (the matrix U ′ with all the entries conjugated) and JV J = V = V t = V * and thus This yields the identity U ′ V = V U ′ .A computation with block multiplication of matrices and the fact that V = V t (along with the facts that U ′ is block diagonal with distinct multiples of the identity matrix along its diagonal) will show that , where V j are n j × n j unitary matrices and Q ℓ and

CONJUGATIONS AND SPECTRAL MEASURES
A version of the spectral theorem for unitary operators [6, Ch.IX, Thm.2.2] (see also [18]) says that if U is a unitary operator on H, then there is a unique spectral measure E(•) on T such that For a spectral measure E(•) and x, y ∈ H, the function defines a finite complex Borel measure on T and, in particular, for each x ∈ H, (4.2) defines a finite positive Borel measure on T, sometimes called an elementary measure.
For a complex-valued Borel measure µ on T, define a new complex Borel measure µ c on the Borel subsets Ω of T by for µ-almost every ξ ∈ T. Therefore, The following proposition, originally explored in [17] for symmetric conjugations, relates a C ∈ C c (U ) with the associated spectral measure From this definition it follows that x,y (Ω) for all x, y ∈ H.
Proposition 4.6.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.
are also spectral measures.Since, for each pair x, y ∈ H, the uniqueness of the spectral measure for a unitary operator gives (a).In a similar way, (b) is a consequence of the computation Note the use of (2.1) in the above calculation.
Proof.For the proof of (a), let x ∈ E(Ω)H and y ∈ (E(Ω)H) ⊥ .By Proposi- For the proof of (b) let we can use Proposition 4.6(d) to see that For a unitary operator U on H with associated spectral measure E(•) and the associated family of elementary measures µ x , x ∈ H from (4.2), one can show, as was done in [21], that for any µ ∈ M + (T) the set is a reducing subspace of U .The space H µ was discussed in [18, §65] as part of a general discussion of the spectral multiplicity theory for unitary operators.
Theorem 4.9.Let U be a unitary operator on H, E(•) its associated spectral measure, µ ∈ M + (T), and C ∈ C c (U ).Then we have the following.
, and thus Recall [18, §48] the standard Boolean operations ∧ and ∨ for µ 1 , µ 2 ∈ M + (T) defined on Borel subsets Ω of T by A is a Borel set}.For a unitary operator U , there exists a scalar spectral measure ν, meaning that ν(∆) = 0 if and only if E(∆) = 0, where E(•) is the spectral measure for U [6, p. 293] (also see the discussion in Theorem 3.8 in [21]).For ν 1 , ν 2 ∈ M + (T) it was shown in [21,Prop. 3.10] Corollary 4.10.Let U be a unitary operator on H and ν be any scalar spectral measure for U .Suppose that Corollary 4.11.Let U be a unitary operator on H and ν be any scalar spectral measure for U .Fix a µ ∈ M + (T).
Proof.By Theorem 4.9 we have CH µ = H µ c ⊆ H µ .Thus, by [21,Prop. 3.11], we obtain Since a unitary operator is normal, we see that ker(U − αI) = ker(U * − ᾱI) i.e., H δα = H δ c α , where δ ξ denotes an atomic measure with atom at ξ ∈ T. This gives us the following corollary.In the above discussion, we often have the hypothesis that C c (U ) = ∅.
As mentioned earlier, this is not always the case (e.g., if U is not unitarily equivalent to its adjoint -Lemma 2.9).

NATURAL CONJUGATIONS ON VECTOR
This section provides a model for conjugations on vector valued Lebesgue spaces and will be useful in our description of C c (U ) in Theorem 8.4.This notation also sets up our discussion of models for bilateral shifts in the next section.
For a Hilbert space H with norm • H and a µ ∈ M + (T), consider the set L 0 (µ, H) of H-valued µ-measurable functions f on T and the set This is often described using tensor notation as L 2 (µ) ⊗ H.
) to denote the scalar valued µ-essentially bounded functions on T. For ease of notation, we will write M ϕ , when ϕ ∈ L ∞ (µ), instead of the more cumbersome M ϕI H , that is, for f ∈ L 2 (µ, H) and µ-almost every ξ ∈ T. The case when ϕ(ξ) = ξ will play an prominent role in this paper in which case we have the vectorvalued bilateral shift M ξ on L 2 (µ, H).
Recall from §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 For any conjugation J on H, define the conjugation J on L 2 (µ, H) by Notice that JM ξ J = Mξ [21].
We now focus our attention on the scalar valued L 2 (µ) space and the set C c (M ξ ).This next result shows that when C c (M ξ ) = ∅, there must be some restrictions on µ.The set C c (M ξ ) was explored in [5] when µ = m.The above (and the antilinearity of C) shows that Therefore, by the weak- * density of the trigonometric polynomials in L ∞ (µ), we obtain where If µ c were not absolutely continuous with respect to µ, then there would be a Borel set Ω ⊆ T such that µ(Ω) = 0 but µ c (Ω) = 0.However, (5.4) leads to contradiction with ϕ = χ Ω , since M χ Ω * = 0 but CM χ Ω C is not.
Remark 5.5.One can adapt the proof of the above proposition to the vectorvalued space L 2 (µ, H).Now let us focus on the situation when µ c ≪ µ.In this case we also have that µ ≪ µ c (Proposition 4.4).For f ∈ L 2 (µ, H) and U ∈ L ∞ (µ, B(H)), it makes sense to write f( ξ) or U( ξ) and define (5.8) for µ-almost every ξ ∈ T. Then we have the following.
(a) J # is a conjugation on L 2 (µ, H); Since J is antilinear on H, one sees that J # is antilinear on L 2 (µ, H).Moreover, for f ∈ L 2 (µ, H) we have Note the use of (4.5) above.Thus, J # is isometric on L 2 (µ, H).
Next we show that (J # ) 2 = I.Indeed, for each f ∈ L 2 (µ, H), Again, note the use of (4.5) above.Therefore, J # is a conjugation.To prove (b), observe that for each f ∈ L 2 (µ, H) we have Then J # is a conjugation on L 2 (µ) and J # M ξ J # = M ξ .
In particular, observe that The following echos a result from [5,Proposition 4.2].Recall the notation from (5.6).
Proposition 5.12.Let J be a conjugation on H, J # be defined by (5.8), and let U ∈ L ∞ (µ, B(H)) be a unitary operator valued function.Then we have the following.
for µ-almost every ξ ∈ T; Proof.For every f ∈ L 2 (µ, H), observe that for µ almost every ξ ∈ T we have Note the use of (5.6) above.This proves (a).
Note that M U J # is antilinear and isometric on L 2 (µ, H).To prove that M U J # is a conjugation (and thus complete the proof of (b)), Lemma 2.7 says that we just need to check the identity By (a) this is equivalent to JU( ξ)J = U * (ξ) since, by (5.1), Statement (c) follows from the fact that M U J # is a conjugation on L 2 (µ, H), and so (M U J # )(M U J # ) = I, along with the fact M U M U * = M U * M U = I (since U(ξ) is unitary for µ-almost every ξ ∈ T).
To see (d), observe that for any f ∈ L 2 (µ, H), which completes the proof of (d).

CONJUGATIONS AND BILATERAL SHIFTS
Many interesting, and naturally occurring, unitary operators are bilateral shifts.Examples include (i) the translation operator iii) the Fourier transform on L 2 (R), (iv) the Hilbert transform on L 2 (R), and (v) the special class of multiplication operators U f = ψf on L 2 (m, T), where ψ is an inner function.We refer the reader to [22,Example 6.3] to see the bilateral nature of each of these operators is worked out carefully.This section gives an initial description of C c (U ) for this class of operators, along with the important fact that C c (U ) = ∅.Another, more concrete, description will be discussed in the next section.Let us begin with a precise definition of 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 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 M ξ on L 2 (m, T) discussed earlier where a wandering subspace M can be taken to be the constant functions.
Though the wandering subspace M in Definition 6.1 is not unique, its dimension is [19].The term "bilateral shift " comes from the fact that since (6.2) every x ∈ H can be uniquely represented as This allows us to define a natural unitary operator Moreover, thanks to (6.2) and (6.3), W U W * = M ξ , where M ξ is the bilateral shift from (5.2) defined on L 2 (m, M) by Note that m = m c and the definition of J # from (5.8) coincides with the one appearing in [5].(b) For a conjugation C on H, the following are equivalent: (ii) C = W CW * is a conjugation on L 2 (m, M) that satisfies any of the equivalent conditions of Theorem 6.4.
Perhaps one might be a bit unsatisfied with the somewhat vague nature of our current description of C c (U ) for a bilateral shift U .The next section will give a much more concrete characterization.

UNITARY MULTIPLICATION OPERATORS ON L 2 (m, T)
As discussed in [22,Example 5.16] there is model for any bilateral shift U on H (recall Definition 6.1) as the multiplication operator M ψ on L 2 = L 2 (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 C c (M ψ ).If J # is the conjugation on L 2 defined by and C ∈ C c (M ψ ), then CJ # is a unitary operator on L 2 for which This trick was used in several places [4,5,8].The bounded operators on L 2 which commute with M ψ , i.e., the commutant of M ψ , were described in [21,Theorem 7.3].
Recall the known fact (see for example [22,Proposition 5.17]) that for an inner function ψ we have the following orthogonal decomposition for L 2 , namely, where K ψ := H 2 ∩ (ψH 2 ) ⊥ is the model space associated with ψ (see [11] for a review of model spaces).In other words, K ψ is a wandering subspace (as in Definition 6.1) for the multiplication operator M ψ .
Let us set up some notation to be used below.For an inner function ψ, let and {h j } 1 j N be a fixed orthonormal basis for K ψ .There are several "natural" orthonormal bases one can choose [11,Ch. 5].Observe that N is finite if and only if ψ is a finite Blaschke product with N zeros, repeated according to multiplicity [11,Prop. 5.19].Also define

The norm of an
When N = ∞, we need to assume that the sum defining f above is finite.Furthermore, the operator 1 j N M ξ (called the inflation of the bilateral shift M ξ on L 2 ) is given by We also define When N = ∞, this is the familiar sequence space ℓ 2 .Finally, observe that (7.2) As a consequence, using the discussion from §5, note that We will actually describe C c (M ψ ) below.
From [21] we have the unitary operator where is a unique decomposition given by [21,Lemma 7.3].Note that and the coefficients a mj arise from the decomposition from (7.1) which yields the unique decomposition Also recall from [21,Thm. 7.3] that Let J and J # denote the standard conjugations on L 2 defined by For our inner function ψ, observe that ψ # = J # ψ is also inner.
Proposition 7.7.For an inner function ψ we have the following. (a) Proof.Part (a) was shown in [4,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 (7.4), where the inner function ψ is replaced by ψ # and orthonormal basis and the orthonormal basis {h j } 1 j N is replaced by the orthonormal basis {h # j } 1 j N , i.e., W # g = [g j ] t 1 j N , where g = There are the two natural conjugations J and J # on 1 j N L 2 defined for each Proposition 7.8.Let ψ be an inner function and {h j } 1 j N be an orthonormal basis for K ψ .Then we have the following.
Proof.Let f ∈ L 2 and observe from (7.5) and (7.6) that Hence which proves (a).The above also yields Theorem 7.9.Suppose that ψ is inner and {h j } 1 j N is an orthonormal basis for K ψ .Then we have the following, almost everywhere on T and Proof.Statement (a) follows from (7.3).To prove (b), observe that since W is a unitary operator, then Since the operator The unitary property gives Φ * (ξ)Φ(ξ) = I and the J # -symmetry property gives Φ(ξ) = Φ( ξ) almost everywhere on T. So far, we have shown that if C is a conjugation which commutes with M ψ , then W CW * = M Φ J # , where Φ satisfies the two properties from (7.10).Conversely suppose that N ) satisfies the two conditions from (7.10).The second condition will show that J # M Φ J # = M * Φ and combining this with the first condition will show that Φ is unitary valued almost everywhere.The second property, along with Proposition 5.12 will show that M Φ J # is a conjugation and belongs to C c (M ξ ).By the discussion above, this says that Applying Proposition 7.8 we can verify the formula (7.11).Indeed, for each and this completes the proof.
Example 7.12.Consider the inner function ψ(z) = z.Here the associated unitary operator M ψ is merely the bilateral shift M ξ on L 2 .In this case, K ψ = C (the constant functions).Moreover, ψ # (z) = z and the expansions from Proposition 7.8 are the standard Fourier expansions of an f ∈ L 2 .Theorem 7.9 says that any C ∈ C c (M ξ ) takes the form for some u ∈ L ∞ that is unimodular and satisfies u(ξ) = u( ξ) almost everywhere on T.
Example 7.15.As a specific nontrivial example of a C ∈ C c (M ξ 2 ) we can take , where t = Arg(ξ).

CONJUGATIONS VIA THE SPECTRAL THEOREM
In this section we use the multiplicity theory for unitary operators [6,18] to describe C c (U ).We also prove that C c (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 [6, p. 307 where for i = ∞, 1, 2, 3, . .., ξ is C i -commuting and (c) For each i = ∞, 1, 2, . . .and any conjugation J (i) on H i , there is a unitary operator valued function for µ i almost every ξ ∈ T and Proof.To show (a) =⇒ (c), let and define the conjugation C = ICI * (note the use of Lemma 2.3).Then Let J (i) be any a conjugation on H i .Since µ c i ≪ µ i for i = ∞, 1, 2, . . ., let and define the map J # (i) on L 2 (µ i , H i ) by for µ i almost every ξ ∈ T and f i ∈ L 2 (µ i , H i ).By Proposition 5.7, each of the above maps defines a conjugation on L 2 (µ i , H i ) which satisfies Use these conjugations to define the conjugation J # = J # (i) on L 2 H and observe that The spectral theorem applied to M ξ also yields the commutant [6, p. 307, Theorem 10.20], namely there are Since C J # is unitary, it follows that M U (i) is also unitary and consequently U (i) is a operator valued operator function such that U (i) (ξ) is unitary for µ i almost every ξ ∈ T. Therefore, Since C| L 2 (µ i ,H i ) is a conjugation, it follows that for µ i almost every ξ ∈ T (Proposition 5.12).This completes the proof of (a) =⇒ (c) .
To prove (c) =⇒ (b), it is enough to take The following yields the converse of Lemma 2.9 and thus completes the criterion as to when C c (U ) = ∅.One can verify that and observe that µ c 1 = µ 2 and µ 2 ≪ µ 1 .Moreover, we can write the Radon-Nikodym derivative From here one sees that h(ξ)h( ξ) = 1 on T as demonstrated in Proposition 4.4.
Example 9.2.Let F denote the standard Fourier-Plancherel transform on L 2 (R).It is well known that F is unitary and that σ(F) = {1, i, −1, −i}.Moreover, the Hermite functions {H n } n 0 form an orthonormal basis for L 2 (R) and FH n = (−i) n H n for all n 0, i.e., the Hermite functions form an eigenbasis for F [15,Ch.11].A description of C s (F), the symmetric conjugations, was given in [21,Example 4.3].In this example we work out C c (F), the commuting conjugations for F. We first note that F ∼ = F * (Example 2.4).Thus, C c (F) = ∅ (Corollary 8.5).
To describe C c (F), we proceed as follows.Our discussion so far says that where E α = ker(F − αI).Define a conjugation J on L 2 (R) for which JH n = H n for all n 0 (initially define J on H n by JH n = H n and extend antilinearly to all of L 2 (R)).
with the standard orthonormal basis {e n } n 0 , and Define a conjugation J on ℓ 2 by J(e n ) = e n for all n 0. Since µ c ≪ µ we can define a conjugation J # on L 2 (µ, ℓ 2 ) such that by (5.8).In other words, with respect to the orthogonal decomposition the conjugation J # can be written in matrix form as The conjugation V * J # V commutes with F and it can be written with respect to the Hermite basis as Therefore, the matrix representation of J # with respect to the orthogonal decomposition in (9.3) is Moreover, by Theorem 8.4, any conjugation C on L 2 (µ, ℓ 2 ) such that C(V FV * ) = (V FV * ) C can be represented by the matrix where U 1 , U −i , U −1 , U i , are unitary operators on ℓ 2 and The first two identities say that the unitary operators U 1 and U −1 are represented by with respect to the basis {e n } n 0 by a matrix with real entries.
The last identity says that the matrix representations in the basis {e n } n 0 of U i and U −i satisfy U −i e m , e n = U * i e m , e n , which we write as U −i = U # i .Therefore, any conjugation C on L 2 (µ, ℓ 2 ) such that C(V FV * ) = (V FV * ) C can be represented as where U R 1 and U R −1 are arbitrary unitary operators on ℓ 2 whose matrix representations with respect to {e n } n 0 have real entries and U i is arbitrary.Finally, a conjugation C on L 2 (R) fulfils the condition CF = FC if and only if it is represented with respect to the decomposition in (9.3) as where U R 1 , U R −1 are arbitrary unitary operators on their respective eigenspaces ker(F − I) and ker(F + I) which are represented in terms of the basis {H 4n } n 0 and {H 4n+2 } n 0 by real matrices, U i is an arbitrary unitary operator on ker(F + iI) and U # i is the unitary operator on ker(F − iI) defined by and E −i has orthonormal basis See [15,Ch. 12] for details.In this example we will describe C c (H ).Note first that C c (H ) = ∅ (recall Example 2.4 and thus H ∼ = H * ).
Similarly as in Example 9.2 we can identify L 2 (R) with L 2 (µ, ℓ 2 ), where µ = δ i +δ −i .Then, the conjugation J # given by equality (5.8) is an antilinear extension of operator J # f n = g n , J # g n = f n , n 1. Putting this in matrix form where J is a conjugation on L 2 (R) which fixes all elements of B i and B −i .
Moreover, by Theorem 8.4, any conjugation C on L 2 (R) with CH C = H * must take the (block) form where U i is arbitrary unitary operator on E i and U # i ∈ B(E −i ) defined as U # i g m , g n = U * i f m , f n similar to the previous example.

A REMARK ABOUT INVARIANT SUBSPACES
The first paper in this series [21] classified, for a fixed unitary operator U on H, the subspaces M of H for which CM ⊆ M for every C s (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 ∈ C c (U ) (the commuting conjugations for U )? We have some partial results in this paper (see for example Proposition 4.8 and Corollary 4.10).However, we do not have a complete and concise characterization.These subspaces seem complicated to describe in the general abstract situation.However, we do have a characterization in the special case where U = M ξ on L 2 (µ, H).Recall the notation from Proposition 5.7.

Lemma 2 . 3 .Example 2 . 4 .
Suppose H and K are Hilbert spaces and V : H → K is a unitary operator.If C is a conjugation on H then V CV * is a conjugation on K.We have already discussed the how the mapping (Cf )(ξ) = f (ξ) on L 2 (m, T) is a conjugation that commutes with the bilateral shift(U f )(ξ) = ξf (ξ).Here are a few other examples.(a)The conjugation (Cf )(x) = f (x) on L 2 (R) commutes with the unitary operator (U f )(x) = f (x − 1).This conjugation also commutes with the Hilbert transform.(b) The conjugation (Cf )(x) = f (−x) on L 2 (R) commutes with the Fourier-Plancherel transform.

Remark 5 . 9 .Corollary 5 . 10 .
If µ = m, Lebesgue measure on T, then m = m c and h ≡ 1 and the conjugation (5.8) coincides with the one considered in[5].A special case worth pointing out is the scalar case H = C.Let µ ∈ M + (T) such that µ c ≪ µ.Let h = dµ c /dµ and define(5.11)

For a bilateralTheorem 6 . 4 .
shift U on H we wish to describe C c (U ).Since W * U W = M ξ on L 2 (m, M), we know that for any C ∈ C c (U ) the mapping C = W * CW is a conjugation on L 2 (m, M) such that CM ξ C = M ξ .The following result from [5, Theorem 4.3] describes C. For M ξ on L 2 (m, M) we have the following.(a) C c (M ξ ) = ∅.(b) Fix a conjugation J on M. For a conjugation C on L 2 (m, M), the following are equivalent.

Theorem 6 .Corollary 6 . 5 .
4 yields a description of C c (U ) when U is a bilateral shift.Let U be a unitary bilateral shift on H with an associated wandering subspace M. The we have the following.(a) C c (U ) = ∅.

Theorem 8 . 4 .
Let U be a unitary operator and C be a conjugation on H.With the notation as in Theorem 8.1, assuming that µ c i ≪ µ i for i = ∞, 1, 2, . . ., the following are equivalent (a) C ∈ C c (U ); (b) For each i = ∞, 1, 2, . . ., there are conjugations

Corollary 8 . 5 . 9 . 1 .
If U is a unitary operator on H such that U ∼ = U * , then there is conjugation C on H such that CU C = U .Proof.If U ∼ = U * , then, as in the proof of Theorem 8.3, the measures µ i and µ c i are mutually absolutely continuous for all i = ∞, 1, 2, . . . .Now invoke Theorem 8.4 with any conjugation J (i) on H j (and U (i) = I H i ) and observe that the conjugationC = I * J # I = I * J # (i) Icommutes with U .9. EXAMPLESExample In Example 7.12 we worked out C c (M ξ ) for the bilaterai shift M ξ on L 2 (m, T).This example contains a description of C c (U ) when U = M ξ on a more complicated L 2 (µ, T) space.Let g : [−1, 1] → [0, ∞) be defined piecewise by g(t) =

Theorem 10 . 1 . 1 . 10 . 2 .
Suppose µ ∈ M + (T) such that µ c ≪ µ and H is a Hilbert space.For a subspace K of L 2 (µ, H) the following are equivalent.(a)CK ⊆ K for every C ∈ C c (M ξ );(b) For a fixed conjugation J on H and J # defined as in (5.8),K is invariant for J # and every M F , where F belongs toL ∞ c (µ, B(H)) := {F ∈ L ∞ (µ, B(H)) : JF(ξ)J = F(ξ) # for µ-a.e.ξ ∈ T}.The proof of this theorem requires a decomposition theorem from [25, proof ofCorollary 3.19].We include a proof for completeness and since the form of the decomposition is important for the proof of Theorem 10.Lemma Any A ∈ B(H) can be expressed as a positive constant times the sum of four unitary operators on H.Proof. DefineH = 1 2 A (A + A * ) and K = 1 2i A (A − A * )

which only one or perhaps both might not appear depending whether 1 or −1 are eigenvalues of U ), and J is the conjugation on C n from (3.2).
[21,t U be a unitary operator with a spectral measure E(•).As previously observed in Proposition 4.6(a), E c (•) is a spectral measure for U * .In[21, Theorem 8.1], the measures µ ∞ , µ 1 , µ 2 , ...fromTheorem 8.1 were constructed using the spectral measure E(•).Therefore, the appropriate measures for operatorU * are µ c ∞ , µ c 1 , µ c 2 ,. . . .We now consider unitary operators with commuting conjugations.Recall from Proposition 5.3 that not all unitary operators have commuting conjugations.In the next result, we re-emphasize this observation in terms of the multiplicity theory from Theorem 8.1.Let U be a unitary operator on H with the multiplicity representation of U given by the mutually singular measures µ ∞ , µ 1 , µ 2 , . . .as in Theorem 8.1.If C c (U ) = ∅, then µ c i ≪ µ i for all i = ∞, 1, 2, . . . .Proof.Lemma 2.9 says that if C c (U ) = ∅, then U ∼ = U * .Hence, by Remark 8.2 and [6, p. 305, Theorem IX 10.16], the measures µ i and µ c i are mutually absolutely continuous for all i = ∞, 1, 2, . . . .We now arrive at the description of C c (U ) in terms of the parameters of the spectral theorem.