Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, and Similarity
Joseph A. Cima, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
aa r X i v : . [ m a t h . F A ] O c t TRUNCATED TOEPLITZ OPERATORS:SPATIAL ISOMORPHISM, UNITARY EQUIVALENCE,AND SIMILARITY
JOSEPH A. CIMA, STEPHAN RAMON GARCIA, WILLIAM T. ROSS,AND WARREN R. WOGEN
Abstract. A truncated Toeplitz operator A ϕ : K Θ → K Θ is the compressionof a Toeplitz operator T ϕ : H → H to a model space K Θ := H ⊖ Θ H .For Θ inner, let T Θ denote the set of all bounded truncated Toeplitz operatorson K Θ . Our main result is a necessary and sufficient condition on inner func-tions Θ and Θ which guarantees that T Θ and T Θ are spatially isomorphic.(i.e., U T Θ = T Θ U for some unitary U : K Θ → K Θ ). We also study op-erators which are unitarily equivalent to truncated Toeplitz operators and weprove that every operator on a finite dimensional Hilbert space is similar to atruncated Toeplitz operator. Introduction
In this paper we consider several questions concerning spatial isomorphism, uni-tary equivalence, and similarity in the setting of truncated Toeplitz operators.Loosely put, a truncated Toeplitz operator is the compression A ϕ : K Θ → K Θ of a standard Toeplitz operator T ϕ : H → H to a Jordan model space K Θ := H ⊖ Θ H (here Θ denotes an inner function). We discuss these definitions andthe related preliminaries in Section 2. The reader is directed to the recent surveyof Sarason [23] for a more thorough account.For a given inner function Θ, we let T Θ denote the set of all bounded truncatedToeplitz operators on K Θ . The main result of the paper (Theorem 3.3) is a simplenecessary and sufficient condition on inner functions Θ and Θ which guaranteesthat the corresponding spaces T Θ and T Θ are spatially isomorphic (i.e., U T Θ = T Θ U for some unitary U : K Θ → K Θ ). This result and its ramifications arediscussed in Section 3 while the proof is presented in Section 4.In Section 5, we study the operators which are unitarily equivalent to truncatedToeplitz operators (UETTO). The class of such operators is surprisingly large andincludes, for instance, the Volterra integration operator [21]. We add to this classby showing that several familiar classes of operators (e.g., normal operators) areUETTO.We conclude this paper in Section 6 by showing that every operator on a finitedimensional Hilbert space is similar to a truncated Toeplitz operator (Theorem6.1). In other words, we prove that the inverse Jordan structure problem is alwayssolvable in the class of truncated Toeplitz operators. This stands in contrast to thesituation for Toeplitz matrices [15]. Preliminaries
In the following, H denotes the classical Hardy space on the open unit disk D [9, 13]. The unit circle | z | = 1 is denoted by ∂ D and we let L := L ( ∂ D ) and L ∞ := L ∞ ( ∂ D ) denote the usual Lebesgue spaces on ∂ D . Model spaces.
To each non-constant inner function Θ there corresponds a modelspace K Θ defined by K Θ := H ⊖ Θ H . (2.1)This terminology stems from the important role that K Θ plays in the model theory for Hilbert space contractions – see [18, Part C].The kernel functions K λ ( z ) = 1 − Θ( λ )Θ( z )1 − λz , z, λ ∈ D , (2.2)belong to K Θ and enjoy the reproducing property h f, K λ i = f ( λ ) , λ ∈ D , f ∈ K Θ . (2.3)If Θ has an angular derivative in the sense of Carath´eodory (ADC) at λ ∈ ∂ D [23, Sect. 2.2] then K λ belongs to K Θ and the formulae (2.2) and (2.3) still hold.Letting P Θ denote the orthogonal projection of L onto K Θ , we observe that[ P Θ f ]( λ ) = h f, K λ i , f ∈ L , λ ∈ D . (2.4)The preceding formula remains valid for λ ∈ ∂ D so long as Θ has an ADC there.We let k λ := K λ k K λ k (2.5)denote the normalized reproducing kernel at λ and, when we wish to be specificabout the underlying inner function Θ involved, we write K Θ λ and k Θ λ in place of K λ and k λ , respectively.There is a natural conjugation (a conjugate-linear isometric involution) on K Θ defined in terms of boundary functions by Cf := f z Θ . (2.6)Although at first glance the expression f z Θ in (2.6) does not appear to correspondto the boundary values of an H function, let alone one in K Θ , a short computationusing (2.1) reveals that if f ∈ K Θ and h ∈ H , then h Cf, Θ h i = 0 = (cid:10) Cf, zh (cid:11) whence Cf indeed belongs to K Θ .A short calculation reveals that[ CK λ ]( z ) = Θ( z ) − Θ( λ ) z − λ . Moreover, the preceding also holds for λ ∈ ∂ D so long as Θ has an ADC there. Truncated Toeplitz operators.
Since K Θ is the closed linear span of the back-ward shifts S ∗ Θ , S ∗ Θ , . . . of Θ [4, p. 83], where S ∗ f = ( f − f (0)) /z , it follows thatthe subspace K ∞ Θ := K Θ ∩ H ∞ of all bounded functions in K Θ is dense in K Θ . RUNCATED TOEPLITZ OPERATORS 3
Keeping these results in mind, for a fixed inner function Θ and any ϕ ∈ L , thecorresponding truncated Toeplitz operator A ϕ : K Θ → K Θ is the densely definedoperator A ϕ f = P Θ ( ϕf ) . (2.7)When we wish to be specific about the underlying inner function Θ, we use thenotation A Θ ϕ to denote the truncated Toeplitz operator with symbol ϕ acting onthe model space K Θ . In most cases, however, Θ is clear from context and we simplywrite A ϕ .Although one can pursue the subject of unbounded truncated Toeplitz operatorsmuch further [24, 25], we are concerned here with those which have a boundedextension to K Θ . Definition 2.8.
Let T Θ denote the set of all truncated Toeplitz operators whichextend boundedly to all of K Θ .Certainly A ϕ ∈ T Θ when ϕ ∈ L ∞ . However [23, Thm. 3.1], there are an abun-dance of unbounded ϕ ∈ L for which A ϕ ∈ T Θ . It is important to note that T Θ is not an algebra since the product of truncated Toeplitz operators need not be atruncated Toeplitz operator (a simple counterexample can easily be deduced from[23, Thm. 5.1]). On the other hand, it turns out that T Θ is a weakly closed linearsubspace of the bounded operators on K Θ [23, Thm. 4.2]. Moreover, if Θ is a finiteBlaschke product of order n , then one can show that dim T Θ = 2 n − Complex symmetric operators.
Of particular importance to the study of trun-cated Toeplitz operators is the notion of a complex symmetric operator [11, 12].Let us briefly discuss the necessary preliminaries. In the following, we let H denotea separable complex Hilbert space and B ( H ) denote the bounded linear operatorson H . Definition 2.9. A conjugation on H is a conjugate-linear operator C : H → H ,which is both involutive (i.e., C = I ) and isometric (i.e., h Cx, Cy i = h y, x i for all x, y ∈ H ) . The standard example of a conjugation is entry-by-entry complex conjugationon an l -space. In fact, each conjugation is unitarily equivalent to the canonicalconjugation on a l -space of the appropriate dimension [11, Lem. 1]. Having dis-cussed conjugations, we next consider certain operators which are compatible withthem. Definition 2.10.
We say that T ∈ B ( H ) is C -symmetric if T ∗ = CT C for someconjugation C on H . We say that T is complex symmetric if there exists a conju-gation C with respect to which T is C -symmetric.Recall the conjugation C defined on K Θ from (2.6). The following result is from[11]. Proposition 2.11.
Every A ϕ ∈ T Θ is C -symmetric. Clark operators.
Let us now review a few necessary facts about the theory ofClark unitary operators [5]. For a more complete account of this theory we refer thereader to [3, 19, 22]. To avoid needless technicalities, we assume that the underlying
J.A. CIMA, S.R. GARCIA, W.T. ROSS, AND W.R. WOGEN inner function Θ satisfies Θ(0) = 0. For α ∈ ∂ D , the operator U α : K Θ → K Θ defined by the formula U α f = A z f + α h f, z Θ i , f ∈ K Θ , (2.12)is called a Clark operator . One can show that each Clark operator U α is unitaryand that every unitary rank-one perturbation of the truncated shift operator A z takes the form U α for some α ∈ ∂ D . Less well-known is the fact that each Clarkoperator U α on K Θ belongs to T Θ [23, p. 524]. There is also the following theorem[23, p. 515]. Theorem 2.13 (Sarason) . If A is a bounded operator on K Θ which commutes with U α for some α ∈ ∂ D , then A ∈ T Θ . Since U α is a cyclic unitary operator [3, Thm. 8.9.10], the Spectral Theoremasserts that there is a measure µ α on ∂ D such that U α is unitarily equivalent tothe operator [ M ζ f ]( ζ ) = ζf ( ζ ) of multiplication by the independent variable ζ on L ( µ α ). Moreover, the measure µ α is carried by the set E α := (cid:26) ζ ∈ ∂ D : lim r → − Θ( rζ ) = α (cid:27) and is therefore singular with respect to Lebesgue measure on ∂ D . The Clarkmeasure µ α constructed above can also easily be obtained using the Herglotz Rep-resentation Theorem for harmonic functions with positive real part [3, Ch. 9]. Asa consequence of this, one can use the fact that Θ(0) = 0 to see that µ α is aprobability measure.It is important to note that the preceding recipe can essentially be reversed. Werecord this observation here for future reference (see [3, p. 202] for details). Proposition 2.14. If µ is a singular probability measure on ∂ D , then there is aninner function Θ with Θ(0) = 0 such that the Clark measure for Θ at α = 1 is µ .In particular, µ is the spectral measure for the Clark unitary operator U on K Θ . In the finite-dimensional case, the Clark measures µ α can be computed explicitly.If Θ is a finite Blaschke product of order n , then dim K Θ = n and the set E α consistsof the n distinct points ζ , ζ , . . . , ζ n on ∂ D for which Θ( ζ j ) = α . The correspondingnormalized reproducing kernels k z j satisfy U α k ζ j = ζ j k ζ j for j = 1 , , . . . , n andform an orthonormal basis for the model space K Θ . Rank one operators in T Θ . Let us conclude these preliminaries with a few wordsconcerning truncated Toeplitz operators of rank one. First recall that for each pair f, g of vectors in a Hilbert space H the operator f ⊗ g : H → H is defined by setting( f ⊗ g )( h ) := h h, g i f. (2.15)Observe that f ⊗ g has a rank one and range C f . Moreover, we also have k f ⊗ g k = k f k k g k . The proof of the next lemma is elementary and is left to the reader. Lemma 2.16.
Let H , H be Hilbert spaces and let f , g ∈ H and f , g ∈ H beunit vectors. (i) If U : H → H is a unitary operator such that U ( f ⊗ g ) U ∗ = f ⊗ g , then there exists a ζ ∈ ∂ D such that U f = ζf and U g = ζg . Inparticular, we have h f , g i H = h f , g i H . RUNCATED TOEPLITZ OPERATORS 5 (ii)
Conversely, if h f , g i = h f , g i , then the operators f ⊗ g and f ⊗ g are unitarily equivalent. The following useful lemma completely characterizes the truncated Toeplitz op-erators of rank one [23, Thm. 5.1, 7.1]. We remind the reader that K λ denotes thereproducing kernel (2.2) for K Θ and C denotes the conjugation on K Θ from (2.6). Lemma 2.17 (Sarason) . Let Θ be an inner function. (i) For each λ ∈ D , the operators K λ ⊗ CK λ and CK λ ⊗ K λ belong to T Θ . (ii) If η ∈ ∂ D and Θ has a ADC at η , then K η ⊗ K η ∈ T Θ . (iii) The only rank-one operators in T Θ are the nonzero scalar multiples of theoperators from (i) and (ii). (iv) If Θ is a Blaschke product of order n , then (a) dim T Θ = 2 n − . (b) If λ , λ , . . . , λ n − are distinct points of D , then the operators K λ i ⊗ CK λ i , ≤ i ≤ n − , form a basis for T Θ . Elementary complex analysis tells us that the automorphism group Aut( D ) of D can be explicitly presented asAut( D ) = { ζϕ a : ζ ∈ ∂ D , a ∈ D } where ϕ a denotes the M¨obius transformation ϕ a ( z ) := z − a − az . (2.18)For an inner function Θ with Θ / ∈ Aut( D ) we have the following lemma: Lemma 2.19.
Suppose Θ is inner with Θ Aut( D ) . (i) If λ , λ ∈ D , then K λ is not a scalar multiple of CK λ . (ii) If also λ = λ , then K λ is not a scalar multiple of K λ . (iii) If λ ∈ D , then K λ ⊗ CK λ is not self-adjoint.Proof. Statements (i) and (ii) are easy computations. For (iii), note that (i) showsthat K λ ⊗ CK λ and ( K λ ⊗ CK λ ) ∗ = CK λ ⊗ K λ have different ranges. (cid:3) When are T Θ and T Θ spatially isomorphic? In this section we consider the problem of determining when two spaces T Θ , T Θ of truncated Toeplitz operators are spatially isomorphic. Let us recall the followingdefinition. Definition 3.1.
For j = 1 ,
2, let H j be a Hilbert space and S j be a subspace of B ( H j ). We say that S is spatially isomorphic to S , written S ∼ = S , if there is aunitary operator U : H → H so that the map S U SU ∗ , S ∈ S , carries S onto S . In this case we often write U S U ∗ = S . J.A. CIMA, S.R. GARCIA, W.T. ROSS, AND W.R. WOGEN
Let us be more precise about our main problem. Spatial isomorphisms of thespaces T Θ give rise to an equivalence relation on the collection of all inner functionsand we wish to determine the structure of the corresponding equivalence classes.If Θ is an inner function and ψ belongs to Aut( D ), then the functions ψ ◦ Θ andΘ ◦ ψ are also inner and hence O (Θ) := { ψ ◦ Θ ◦ ψ : ψ , ψ ∈ Aut( D ) } consists precisely of those inner functions that can be obtained from Θ by pre- andpost-composition with disk automorphisms. It turns out that while Θ ∈ O (Θ ) isa sufficient condition for ensuring that T Θ ∼ = T Θ , it is not necessary. To formulatethe correct theorem, we introduce the conjugation f f on H by setting f ( z ) := f ( z ) (3.2)and we note that Θ is inner if and only if Θ is. Moreover, note that the L . The main theorem ofthis section is the following: Theorem 3.3.
For inner functions Θ and Θ , T Θ ∼ = T Θ ⇔ Θ ∈ O (Θ ) ∪ O (Θ ) . (3.4)The proof of the preceding theorem is somewhat long and it requires a numberof technical lemmas. We therefore defer the proof until Section 4.It is natural to ask if there are simple geometric conditions on the zeros of Blashkeproducts B and B that will ensure that T B ∼ = T B . While the general questionappears difficult, several partial results are available. For instance, if B and B areBlaschke products of order 2, then T B ∼ = T B (see Theorem 5.2 below). Anotherspecial case is handled by the following corollary. Before presenting it, we requirea few words concerning the hyperbolic metric on D .The hyperbolic (or Poincar´e) metric on D is defined for z , z ∈ D by ρ ( z , z ) = inf γ Z γ | dz | − | z | , (3.5)where the infimum is taken over all arcs γ in D connecting z and z . It is well-known that the hyperbolic metric ρ is conformally invariant in the sense that ρ ( z , z ) = ρ ( ψ ( z ) , ψ ( z )) , ∀ ψ ∈ Aut( D ) . Moreover, ρ (0 , z ) = log 1 + | z | − | z | (3.6)and the geodesic through 0 , z turns out to be [0 , z ], the line segment from 0 to z .The reader can consult [13, p. 4] for further details. Corollary 3.7.
For a finite Blaschke product B of order n , we have T z n ∼ = T B ifand only if either B has one zero of order n or B has n distinct zeros all lying on acircle Γ in D with the property that if these zeros are ordered according to increasingargument on Γ , then adjacent zeros are equidistant in the hyperbolic metric (3.5) .Proof. Suppose that T z n ∼ = T B . Noting that ( z n ) = z n and applying Theorem 3.3we conclude that B = ψ ◦ ϕ n for some ϕ, ψ ∈ Aut( D ). If ψ is a rotation then B RUNCATED TOEPLITZ OPERATORS 7 has one zero of order n . If ψ is not a rotation, then the zeros z , z , . . . , z n of B aredistinct and satisfy the equation (cid:18) z j − a − az j (cid:19) n = b for some a, b ∈ D . The n th roots of b are equally spaced on a circle of radius | b | centered at the origin (which are also equally spaced with respect to the hyperbolicmetric). The z j are formed by applying a disk automorphism to these n th roots of b and thus, by the conformal invariance of the hyperbolic metric, are equally spaced(in the hyperbolic metric) points on some circle Γ in D .Now assume that the zeros z , z , . . . , z n of B satisfy the hypothesis above. If z = z = · · · = z n , then B is the n th power of a disk automorphism and hencebelongs to O ( z n ). In this case, we conclude that T B ∼ = T z n . In the second case,map the hyperbolic center of the circle Γ to the origin with a disk automorphism ψ .The map ψ will also map the circle Γ to a circle | z | = r having the same hyperbolicradius as Γ. Consequently, ψ will map the zeros of B to points t , t , . . . , t n on | z | = r which are equally spaced in the hyperbolic metric. By basic properties ofthe hyperbolic metric, these points take the form t j = ω j a where ω is a primitive n th root of unity and a ∈ D . Putting this all together, we get that the zeros z , z , · · · , z n of B satisfy z j = ψ − ( w j a )and hence B = ψ n − a − aψ n ∈ O ( z n ) . By Theorem 3.3 we conclude that T B ∼ = T z n . (cid:3) Remark 3.8.
For any inner function Θ, a well-known theorem of Frostman [13,p. 79] implies there are many ψ ∈ Aut( D ) for which B = ψ ◦ Θ is a Blaschke product.An application of Theorem 3.3 shows that T Θ ∼ = T B . It is natural to ask whetheror not there are infinite Blaschke products B for which T Θ ∼ = T B implies that Θis a Blaschke product. Again, using Theorem 3.3, this can be rephrased as: for afixed infinite Blaschke product B , when does O ( B ) ∪ O ( B ) contain only Blaschkeproducts? A little exercise will show that this is true precisely when ψ ◦ B is aBlaschke product for every ψ ∈ Aut( D ). Blaschke products satisfying this propertyare called indestructible (see [20] and the references therein). It is well-known that Frostman Blaschke products i.e., those Blaschke products B which satisfysup ζ ∈ ∂ D ∞ X n =1 − | a n | | ζ − a n | < ∞ , where ( a n ) n ≥ are the zeros of B , repeated accordingly to multiplicity, are inde-structible. Moreover, using a deep theorem of Hruscev and Vinogradov concerningthe inner multipliers of the space of Cauchy transforms of measures on the unitcircle [3, Ch. 6] along with a result from [17], one can show that O ( B ) ∪ O ( B )contains only Frostman Blaschke products if and only if B is a Frostman Blaschkeproduct. J.A. CIMA, S.R. GARCIA, W.T. ROSS, AND W.R. WOGEN Proof of Theorem 3.3
The proof of Theorem 3.3 is somewhat lengthy and it is consequently broken upinto a series of propositions and lemmas. For the sake of clarity, we deal with theimplications ( ⇐ ) and ( ⇒ ) in equation (3.4) separately. Proof of the implication ( ⇐ ) in (3.4) . This is the simpler portion of the proofand it boils down to several computational results.
Proposition 4.1. If Θ is inner and ψ ∈ Aut( D ) , then T Θ ∼ = T Θ ◦ ψ .Proof. Let ψ ( z ) = η z − a − az , η ∈ ∂ D , a ∈ D , (4.2)be a typical disk automorphism and define U : H → H by U f = p ψ ′ ( f ◦ ψ ) . One can check by the change of variables formula that U is a unitary operator and U − f = U ∗ f = p ( ψ − ) ′ ( f ◦ ψ − ) . Next observe that if f ∈ K Θ , then h U f, (Θ ◦ ψ ) h i = h f, U ∗ ((Θ ◦ ψ ) h ) i = D f, Θ p ( ψ − ) ′ ( h ◦ ψ − ) E = 0for all h ∈ H . Similarly, for g ∈ K Θ ◦ ψ we have h U ∗ g, Θ h i = h g, U (Θ h ) i = D g, p ψ ′ (Θ ◦ ψ )( h ◦ ψ ) E = 0 . Thus U K Θ = K Θ ◦ ψ and hence U restricts to a unitary map from K Θ onto K Θ ◦ ψ ,which we also denote by U .If A Θ g ∈ T Θ , then observe that for f ∈ K Θ we have[ U A Θ g f ]( λ ) = [ U P Θ ( gf )]( λ )= p ψ ′ ( λ )[ P Θ ( gf )]( ψ ( λ ))= p ψ ′ ( λ ) (cid:10) gf, k ψ ( λ ) (cid:11) by (2.4)= p ψ ′ ( λ ) Z ∂ D g ( ζ ) f ( ζ ) − Θ( ψ ( λ ))Θ( ζ )1 − ζψ ( λ ) ! | dζ | π Now make the change of variables ζ = ψ ( w ) and use the identities ψ ( z ) = η z − a − az , ψ ′ ( z ) = η − | a | (1 − az ) to show that the above is equal to Z ∂ D p ψ ′ ( w ) g ( ψ ( w )) f ( ψ ( w )) 1 − Θ( ψ ( λ ))Θ( ψ ( w ))1 − wλ | dw | π = [ A Θ ◦ ψg ◦ ψ U f ]( λ ) . From this we conclude that
U A Θ g = A Θ ◦ ψg ◦ ψ U whence A U AU ∗ is a spatial iso-morphism between T Θ and T Θ ◦ ψ . (cid:3) The computational portion of the following proposition is originally due to Cro-foot [7]. A detailed discussion of these so-called
Crofoot transforms in the contextof truncated Toeplitz operators can be found in [23, Sec. 13].
RUNCATED TOEPLITZ OPERATORS 9
Proposition 4.3 (Crofoot) . If Θ is inner, a ∈ D , and ϕ a denotes the M¨obiustransformation (2.18) , then U f := p − | a | − a Θ f defines a unitary operator from K Θ to K ϕ a ◦ Θ . Moreover, U T Θ U ∗ = T ϕ a ◦ Θ . Thusfor any ψ ∈ Aut( D ) we have T Θ ∼ = T ψ ◦ Θ . Our next goal is to establish that T Θ ∼ = T Θ . This is the content of Proposition4.6 below. We should remark that this observation is closely related to [1, Cor. 1.7,Prop. 1.8]. The proof of Proposition 4.6 requires two preliminary lemmas. First,recall the definitions (2.6) of the conjugation C on the model space K Θ and (3.2) ofthe conjugation f f . Now let C denote the corresponding conjugation on themodel space K Θ . Finally, we define a conjugate-linear map J on K Θ by Jf = f . Lemma 4.4.
For Θ inner, (i) J K Θ = K Θ . (ii) If g ∈ K Θ , then J − g = g . (iii) JC : K Θ → K Θ is unitary. Also, the following formulae hold JC = C J, ( JC ) ∗ = CJ − = J − C . (iv) For all λ ∈ D , we have JCK Θ λ = C K Θ λ , JC ( CK Θ λ ) = K Θ λ . Proof.
Statement (i) follows from the fact that f f is a conjugation on H andhence 0 = h f, Θ h i = (cid:10) Θ h , f (cid:11) , f ∈ K Θ , h ∈ H . Statement (ii) is immediate since f → f is an involution on H . For (iii), it is clearthat JC is unitary since J and C are isometric and conjugate linear. The remainingidentities in (iii) can be easily checked. For (iv) first compute JK Θ λ = K Θ λ andfinish by using JC = C J . (cid:3) Lemma 4.5. If A Θ ϕ ∈ T Θ , then JA Θ ϕ J − = A Θ ϕ .Proof. For all f, g ∈ K ∞ Θ , (cid:10) JA Θ ϕ J − f, g (cid:11) = (cid:10) Jg, A Θ ϕ J − f (cid:11) = (cid:10) ( A Θ ϕ ) ∗ Jg, J − f (cid:11) = Z π ϕ ( e iθ ) g ( e − iθ ) f ( e − iθ ) dθ π = Z − π ϕ ( e − iθ ) g ( e iθ ) f ( e iθ ) − dθ π = Z π ϕ ( e − iθ ) g ( e iθ ) f ( e iθ ) dθ π = (cid:10) ϕ f, g (cid:11) = D A Θ ϕ f, g E . (cid:3) Armed now with Lemmas 4.4 and 4.5 we are ready to prove the following.
Proposition 4.6.
For Θ inner, T Θ ∼ = T Θ .Proof. From Lemma 4.4, the operator JC : K Θ → K Θ , (4.7)is unitary. Furthermore, for f, g ∈ K Θ we have (cid:10) ( JC ) A Θ ϕ ( JC ) ∗ f, g (cid:11) = (cid:10) C JA Θ ϕ J − C f, g (cid:11) (by Lemma 4.4)= D C A Θ ϕ C f, g E (by Lemma 4.5)= D ( A Θ ϕ ) ∗ f, g E (Proposition 2.11)= D A Θ ϕ f, g E . It follows that A ( JC ) A ( JC ) ∗ is a spatial isomorphism from T Θ onto T Θ . (cid:3) Propositions 4.1, 4.3, and 4.6 yield the implication ( ⇐ ) of (3.4). This completesthe first part of the proof of Theorem 3.3. Technical Lemmas.
The proof of the ( ⇒ ) implication in (3.4) is significantly moreinvolved than the proof of ( ⇐ ). We require several additional technical lemmaswhich we present in this subsection. Lemma 4.8.
Let Θ be inner, Θ Aut( D ) , and let L Θ := { ρk λ : ρ ∈ ∂ D , λ ∈ D } , e L Θ := { ρCk λ : ρ ∈ ∂ D , λ ∈ D } . For each fixed λ ∈ D , we have dist (cid:16) k λ , e L Θ (cid:17) > , (4.9)dist ( Ck λ , L Θ ) > . (4.10) Proof.
Suppose that dist (cid:16) k λ , e L Θ (cid:17) = 0 holds for some λ ∈ D . It follows thatthere are sequences ( µ n ) n ≥ ⊂ D and ( ρ n ) n ≥ ⊂ ∂ D so that ρ n Ck µ n → k λ (4.11)in the norm of H . Passing to a subsequence, we can assume that µ n converges tosome µ ∈ D − . There are two cases we must consider. Case 1 : If µ ∈ D , then Ck µ n → Ck µ in H and hence pointwise in D . This forces the sequence ρ n to converge to some ρ ∈ ∂ D and hence k λ = ρ Ck µ . However, this contradicts Lemma 2.19 from which we conclude that µ ∈ ∂ D . Case 2 : If µ ∈ ∂ D , then the sequence Θ( µ n ) is bounded and hence upon passingto a subsequence we may assume that Θ( µ n ) → a for some a ∈ D − . By (4.11) itfollows that ρ n Θ( z ) − Θ( µ n )( z − µ n ) k CK µ n k H −→ − Θ( λ )Θ( z )(1 − λ z ) k K λ k (4.12) RUNCATED TOEPLITZ OPERATORS 11 whence we also have pointwise convergence on D . For any fixed z ∈ D for whichΘ( z ) = a we conclude that ρ n Θ( z ) − Θ( µ n )( z − µ n ) k CK µ n k → − Θ( λ )Θ( z )(1 − λ z ) k K λ k 6 = 0 . But since Θ( z ) − Θ( µ n ) z − µ n → Θ( z ) − az − µ = 0 , it follows that ρ n converges to some ρ ∈ ∂ D and k CK µ n k − converges to somefinite number M . Upon letting n → ∞ in (4.12), we obtain ρ M Θ( z ) − az − µ = 1 − Θ( λ )Θ( z )(1 − λ z ) k K λ k . Solving for Θ( z ) in the preceding reveals that Θ is a linear fractional transformation– contradicting the assumption that Θ Aut( D ). This establishes (4.9). Thesecond inequality (4.10) follows immediately since C is an involutive isometry andso dist( k λ , e L Θ ) = dist( Ck λ , C L Θ ) = dist( Ck λ , L Θ ) . (cid:3) We henceforth assume that Θ and Θ are fixed inner functions, neither inAut( D ), and that U T Θ U ∗ = T Θ for some unitary U : K Θ → K Θ . We let C , C denote the conjugations (2.6) on K Θ and K Θ , respectively. To simplify ournotation somewhat, we set k λ := k Θ λ , e k λ := C k Θ λ , ℓ λ := k Θ λ , e ℓ λ := C k Θ λ for λ ∈ D .We now exploit the fact that the rank-one operators in T Θ are carried onto therank-one operators in T Θ by our spatial isomorphism. By Lemma 2.17 and Lemma2.19, we conclude that U ( k λ ⊗ e k λ ) U ∗ is either ζℓ η ⊗ e ℓ η for some ζ ∈ ∂ D and η ∈ D ,or ζ ′ e ℓ η ′ ⊗ ℓ η ′ for some ζ ′ ∈ ∂ D and η ′ ∈ D . Upon applying Lemma 2.16 we observethat U k λ ∈ L Θ ∪ e L Θ . (4.13)In fact, even more is true. Lemma 4.14.
Either U L Θ = L Θ or U L Θ = e L Θ . As a consequence, there aremaps w : D → ∂ D and ϕ : D → D so that either U ( k λ ⊗ e k λ ) = w ( λ ) ℓ ϕ ( λ ) ⊗ e ℓ ϕ ( λ ) , ∀ λ ∈ D , or U ( k λ ⊗ e k λ ) = w ( λ ) e ℓ ϕ ( λ ) ⊗ ℓ ϕ ( λ ) , ∀ λ ∈ D . Proof.
Since the map λ k λ is continuous from D to K Θ , it follows that F ( λ ) := U k λ is a continuous function from D to K Θ . Suppose that F ( λ ) = ρ ℓ η ∈ L Θ forsome λ , η ∈ D , ρ ∈ ∂ D . We now show that there is an open disk B ( λ , δ ) about λ (of radius δ >
0) so that λ ∈ B ( λ , δ ) ⇒ U k λ ∈ L Θ . If this were not the case then by (4.13) there exists sequences λ n → λ , η n ∈ D , ρ n ∈ ∂ D so that F ( λ n ) = ρ n e ℓ η n . By the continuity of F at λ , we see that ρ n e ℓ η n → ρ ℓ η ,which contradicts Lemma 4.8. Since D is connected, we conclude that U L Θ ⊂ L Θ .If we now interchange the roles of Θ and Θ , replacing U with U ∗ , the argumentabove shows that U ∗ L Θ ⊂ L Θ . This means that L Θ ⊂ U L Θ and so U L Θ = L Θ . The same argument shows that if F ( λ ) ∈ e L Θ , then U L Θ = e L Θ . (cid:3) Remark 4.15.
Now observe that it suffices to consider the case where U L Θ = L Θ . Indeed, suppose that U L Θ = e L Θ . We know from Proposition 4.6 that T Θ ∼ = T Θ and, from Lemma 4.4 part (iv), the unitary JC implementing thisspatial isomorphism carries e L Θ onto L Θ . By replacing Θ with Θ if necessary(which does not change O (Θ ) ∪ O (Θ )), we assume for the remainder of the proofthat U L Θ = L Θ . Under this assumption it follows that U ( k λ ⊗ e k λ ) = w ( λ ) ℓ ϕ ( λ ) ⊗ e ℓ ϕ ( λ ) , ∀ λ ∈ D , (4.16)for some functions w : D → ∂ D and ϕ : D → D . Lemma 4.17.
The function ϕ in (4.16) belongs to Aut( D ) .Proof. We first prove that ϕ : D → D is a bijection. Suppose that ϕ ( λ ) = ϕ ( λ ). Itfollows from (4.16) that k λ = ck λ for some scalar c . By Lemma 2.19, we concludethat λ = λ whence ϕ is injective. Now let η ∈ D . By Lemma 4.14 we know that U ∗ ( ℓ η ⊗ e ℓ η ) U = ck λ ⊗ e k λ for some λ ∈ D and some scalar c . We cannot have U ∗ ( ℓ η ⊗ e ℓ η ) U = c e k λ ⊗ k λ or else (by Lemma (2.16)) U k λ = c e ℓ η which we are assuming is not the case.Another application of Lemma 2.16 reveals that ϕ ( λ ) = η whence ϕ is surjective.To show that ϕ ∈ Aut( D ), it suffices to prove that ϕ is analytic on D . Wemay assume that Θ (0) = 0 and Θ ( w ) = 0 for some w ∈ D . If this is not thecase, choose a , a ∈ D ( a = a ) so that Θ ( a ) = Θ ( a ) = b , replace Θ by ϕ b ◦ Θ ◦ ϕ − a , and appeal to Propositions 4.1 and 4.3. In particular, this meansthat if L η denotes the reproducing kernel for K Θ , then L = 1 , L w = 11 − w z . Let f = U − L and g = U − L w . Then for any λ ∈ D we have f ( λ ) = h f, K λ i = h U f, U K λ i = * , w ( λ ) k K λ k (cid:13)(cid:13) L ϕ ( λ ) (cid:13)(cid:13) L ϕ ( λ ) + = w ( λ ) k K λ k (cid:13)(cid:13) L ϕ ( λ ) (cid:13)(cid:13) . Similarly, using the formula for f ( λ ) above, we get g ( λ ) = h g, K λ i = h U g, U K λ i RUNCATED TOEPLITZ OPERATORS 13 = * − w z , w ( λ ) k K λ k (cid:13)(cid:13) L ϕ ( λ ) (cid:13)(cid:13) L ϕ ( λ ) + = 11 − w ϕ ( λ ) w ( λ ) k K λ k (cid:13)(cid:13) L ϕ ( λ ) (cid:13)(cid:13) = 11 − w ϕ ( λ ) f ( λ ) . Since the functions f and g are analytic (and not identically zero) on D , uponsolving for ϕ ( λ ) in the preceding identity we conclude that ϕ is analytic on D . (cid:3) Proof of the implication ( ⇒ ) in (3.4) . We have already seen via Propositions4.1, 4.3, and 4.6 thatΘ ∈ O (Θ ) ∪ O (Θ ) ⇒ T Θ ∼ = T Θ . We now prove the reverse implication. In light of Remark 4.15 and Lemma 4.17 wemay assume that
U k λ = w ( λ ) ℓ ϕ ( λ ) , ∀ λ ∈ D , (4.18)for some functions w : D → ∂ D and ϕ ∈ Aut( D ). Consequently we may appeal toLemma 2.16 to conclude that U (cid:16) k λ ⊗ e k λ (cid:17) U ∗ = w ( λ ) ℓ ϕ ( λ ) ⊗ e ℓ ϕ ( λ ) . Upon taking adjoints in the preceding equation we then obtain U (cid:16)e k λ ⊗ k λ (cid:17) U ∗ = w ( λ ) e ℓ ϕ ( λ ) ⊗ ℓ ϕ ( λ ) . Lemma 2.16 now yields U e k λ = w ( λ ) e ℓ ϕ ( λ ) . (4.19)Next we combine (4.18) and (4.19) to obtain |h e k λ , k λ i| = |h e ℓ ϕ ( λ ) , ℓ ϕ ( λ ) i| . Noting that k λ = K λ k K λ k , h CK λ , K λ i = Θ ′ ( λ ) , k CK λ k = k K λ k = s − | Θ( λ ) | − | λ | we get | Θ ′ ( λ ) | (1 − | λ | )1 − | Θ ( λ ) | = | Θ ′ ( ϕ ( λ )) | (1 − | ϕ ( λ ) | )1 − | Θ ( ϕ ( λ )) | . (4.20)Using the Schwarz-Pick lemma [13, p. 2] we have | ϕ ′ ( z ) | = 1 − | ϕ ( z ) | − | z | , ∀ z ∈ D , ∀ ϕ ∈ Aut( D ) , whence the identity (4.20) becomes | Θ ′ ( λ ) | − | Θ ( λ ) | = | Θ ′ ( ϕ ( λ )) | − | Θ ( ϕ ( λ )) | | ϕ ′ ( λ ) | . Replacing Θ by Θ ◦ ϕ in the preceding formula gives us | Θ ′ ( λ ) | − | Θ ( λ ) | = | Θ ′ ( λ ) | − | Θ ( λ ) | . (4.21) Another application of the Schwarz-Pick lemma shows that (4.21) continues to holdif Θ is replaced by ψ ◦ Θ for all ψ ∈ Aut( D ). It follows that we may assume thatΘ (0) = Θ (0) = 0 , Θ ′ (0) = 0 , Θ ′ (0) = 0 . (4.22)If not, choose a ∈ D so that Θ ′ ( a ) = 0 and Θ ′ ( a ) = 0. Let b = Θ ( ϕ − a (0)) and b = Θ ( ϕ − a (0)). Now replace Θ by ϕ b ◦ Θ ◦ ϕ − a and Θ by ϕ b ◦ Θ ◦ ϕ − a andobserve that (4.22) still holds. It is important to note that all of these simplifyingassumptions on Θ has not altered O (Θ ) ∪ O (Θ ).The assumption (4.22) means that both Θ and Θ are invertible near the origin.Thus there is an ε > and Θ are injective on the disk B (0 , ε ). Thereis also a δ > B (0 , δ ) ⊂ Θ ( B (0 , ε )) and B (0 , δ ) ⊂ Θ ( B (0 , ε )).Now suppose that | z | < δ . Then Θ − ([0 , z ]) is a curve γ in B (0 , ε ) and Θ ◦ Θ − ([0 , z ]) = Θ ( γ ) is a curve Γ in B (0 , δ ) going from 0 to β := Θ ◦ Θ − ( z ). Fromour discussion in the previous paragraph along with the change of variables formulaand (4.21) we get Z γ | Θ ′ ( t ) | − | Θ ( t ) | dt = Z γ | Θ ′ ( t ) | − | Θ ( t ) | dt = Z Γ | dw | − | w | . Thus ρ (0 , z ) ≥ ρ (0 , β ) whence, by (3.6), | z | ≥ | β | and so | Θ ◦ Θ − ( z ) | ≤ | z | for small | z | . A similar argument also shows that | Θ ◦ Θ − | ≤ | z | for small | z | .Putting this all together we find that | z | = | Θ ◦ Θ − ( z ) | , ∀| z | < δ and hence there is a ζ ∈ ∂ D such thatΘ ◦ Θ − ( z ) = ζz, ∀| z | < δ. Replacing z by Θ ( z ) for | z | small, we have Θ ( z ) = ζ Θ ( z ) and so Θ = ζ Θ on D . Thus Θ ∈ O (Θ ) as desired. This completes the proof of Theorem 3.3. (cid:3) Unitary equivalence to a truncated Toeplitz operator
In this section we attempt to describe those classes of Hilbert space operatorswhich are UETTO (unitarily equivalent to a truncated Toeplitz operator). Thisquestion is more subtle that it might at first appear. For instance, the Volterraintegration operator, being the Cayley transform of the compressed shift A z ona certain model space, is UETTO [21] (see also [18, p. 41]). While the generalquestion appears quite difficult, we are able to obtain concrete results in a fewspecific cases. Theorem 5.1.
Every rank one operator is UETTO.Proof.
Let T = u ⊗ v be a rank one operator on an n -dimensional Hilbert space.Without loss of generality, suppose that 2 ≤ n ≤ ∞ , k u k = k v k = 1 and0 ≤ h u, v i ≤ . We claim that there exists a Blaschke product Θ of order n (i.e., having n zeros,counting according to multiplicity) and an appropriate λ so that u ⊗ v is unitarilyequivalent to a multiple of k λ ⊗ Ck λ . By Lemmas 2.16 and 2.17 it suffices to exhibitΘ and λ so that h u, v i = h k λ , Ck λ i . RUNCATED TOEPLITZ OPERATORS 15
There are three cases to consider:(i) Suppose that h u, v i = 0. In this case let Θ be a Blaschke product of order n having a repeated root at λ = 0. Then h k , Ck i = (cid:28) , Θ z (cid:29) = Θ ′ (0) = 0 = h u, v i as desired.(ii) Suppose that h u, v i = 1. Since u and v are unit vectors, it follows that u = v . In this case, let Θ be a Blaschke product of order n having an ADCat λ = 1 and satisfying Θ(1) = 1 in the non-tangential limiting sense. Ashort computation shows that Ck = k whence h k , Ck i = 1 = h u, v i as desired.(iii) Suppose that 0 < h u, v i <
1. In this case, let Θ be a Blaschke productof order n with a simple root at λ = 0 and having its remaining roots λ i being strictly positive. In this case h k , Ck i = Θ ′ (0) = n Y i =1 λ i . By selecting the zeros λ i appropriately, the preceding can be made to equal h u, v i as was required. (cid:3) Theorem 5.2.
Every × matrix is UETTO. In fact, if T is a given × matrixand Θ is a Blaschke product of order , then T Θ contains an operator unitarilyequivalent to T .Proof. Let T be a given 2 × × T is complex symmetric: T = T t . Now observe that the subspace of S ( C ) ⊂ M ( C ) consisting of all 2 × T Θ = 3 as well. If β is a C -real orthonormal basis for K Θ (see [10, Lem. 2.6] for details), then themap Φ : T Θ → S ( C ) defined by Φ( A ) = [ A ] β is clearly injective whence its imagecontains T [10, Lem. 2.7]. (cid:3) Corollary 5.3. If Θ and Θ are Blaschke products of order 2, then T Θ ∼ = T Θ .Proof. The proof of Theorem 5.2 provides a recipe for constructing spatial isomor-phisms Φ : T Θ → S ( C ) and Φ : T Θ → S ( C ). It follows that Φ ◦ Φ : T Θ →T Θ is a spatial isomorphism. (cid:3) Theorem 5.4. If N is an n × n normal matrix and Θ is a Blaschke product oforder n , then N is unitarily equivalent to an operator in T Θ .Proof. By the Spectral Theorem, we know that N is unitarily equivalent to thediagonal matrix diag( λ , λ , . . . , λ n ) where λ , λ , . . . , λ n denote the eigenvalues of N , repeated according to their multiplicity. Select a Clark unitary operator U = U α (see (2.12)) and note from Theorem 2.13 that U ∈ T Θ as is p ( U ) for any polynomial p ( z ). Also note that the eigenvalues ζ , ζ , . . . , ζ n of U have multiplicity one [5, Thm. 3.2] (see also [10, Thm. 8.2]). Thus, there exists a polynomial p ( z ) suchthat p ( ζ i ) = λ i for i = 1 , , . . . , n . It follows that p ( U ) is unitarily equivalent todiag( λ , λ , . . . , λ n ) and hence to N itself. (cid:3) If we are willing to sacrifice the arbitrary selection of Θ, then the precedingcan be generalized to the infinite-dimensional setting. To do so, we require somepreliminary remarks on multiplication operators. For a compactly supported Borelmeasure µ on C , we have the associated algebra M µ := { M ϕ ∈ B ( L ( µ )) : ϕ ∈ L ∞ ( µ ) } (5.5)of multiplication operators on L ( µ ). For each such measure we define the orderedpair κ ( µ ) = ( ǫ, n ) where ǫ = ( µ is purely atomic , , and 0 ≤ n ≤ ∞ denotes the number of atoms of µ . In terms of the function κ ,the following theorem of Halmos and von Neumann [14] (see also [6, Thm. 7.51.7])describes when the algebras (5.5) are spatially isomorphic. Theorem 5.6 (Halmos and von Neumann) . For two compactly supported Borelmeasures µ , µ on C , the algebras M µ and M µ are spatially isomorphic if andonly if κ ( µ ) = κ ( µ ) . Theorem 5.7.
Every normal operator on a separable Hilbert space is UETTO.Proof. If N is a normal operator on a separable Hilbert space, then the spectraltheorem asserts that N is unitarily equivalent to M ϕ : L ( µ ) → L ( µ ) for somecompactly supported Borel measure µ on C and some ϕ ∈ L ∞ ( µ ). Let η be a singular probability measure on ∂ D for which κ ( µ ) = κ ( η ). By Theorem 5.6, M ϕ : L ( µ ) → L ( µ ) is unitarily equivalent to M ψ : L ( η ) → L ( η ), for some ψ ∈ L ∞ ( η ).By Proposition 2.14, η is a Clark measure for some Clark unitary operator U on K Θ for some inner Θ. Again by Proposition 2.14, U is unitarily equivalent to( M z , L ( η )). Moreover, by Theorem 2.13, we also get that U as well as ψ ( U )belong to T Θ . Finally, note that ψ ( U ) ∼ = ( M ψ , L ( η )) ∼ = ( M ϕ , L ( µ )) ∼ = N. In the previous line we use ∼ = to denote unitary equivalence of two operators. (cid:3) Theorem 5.8.
For k ∈ N ∪ {∞} , the k -fold inflation of a finite Toeplitz matrix isUETTO.Proof. Suppose that n ∈ N and A ψ ∈ T z n , where ψ ( ζ ) = n − X m = − n +1 a m ζ m (5.9)is a trigonometric polynomial. In particular, the matrix of A ψ relative to theusual monomial basis { , z, . . . , z n − } for K z n is a Toeplitz matrix and every finiteToeplitz matrix arises in this manner.For k ∈ N ∪ {∞} let A ψ ⊗ I denote the k -fold inflation of A ψ , where I is theidentity matrix on some k -dimensional Hilbert space. We will now show that A ψ ⊗ I RUNCATED TOEPLITZ OPERATORS 17 is UETTO. To do this let B be a Blaschke product of order k (Note that k can beinfinite). If T B denotes the usual Toeplitz operator on H with symbol B , then T B ( B j K B ) = B j +1 K B , j = 0 , , , . . . . Since H = ∞ M j =0 B j K B , we see that T B is unitarily equivalent to a shift of multiplicity k , i.e., T B ∼ = T z ⊗ I (This is a standard fact from operator theory [6, p. 111]). In a similar way, oneshows that T B m ∼ = T z m ⊗ I, m ∈ Z , and so, from (5.9), T ψ ( B ) ∼ = T ψ ⊗ I. A short exercise using the fact that K B = ( BH ) ⊥ will show that K B n = n − M j =0 B j K B . Combine this with the above discussion to show that A ψ ( B ) : K B n → K B n (whichis the compression of T ψ ( B ) to K B n ) is unitarily equivalent to A ψ ⊗ I . (cid:3) We conclude this section with several open questions. The first two are motivatedby Theorem 5.8.
Question 5.10.
For which truncated Toeplitz operators A Θ ϕ and for which k ∈ N ∪ {∞} is the k -fold inflation of A Θ ϕ UETTO?
Question 5.11.
When is the direct sum of truncated Toeplitz operators UETTO?It is known that every truncated Toeplitz operator is a complex symmetric op-erator (see Definition 2.10 and Proposition 2.11). Moreover, so is the Volterraintegration operator, every 2 × Question 5.12.
Which complex symmetric operators are UETTO?6.
Similarity to a truncated Toeplitz operator
It was asked in [16] whether or not the inverse Jordan problem can be solvedin the class of Toeplitz matrices. That is to say, given any Jordan canonical form,can one find a Toeplitz matrix that is similar to this form? A negative answer tothis question was subsequently provided by G. Heinig [15]. On the other hand, itturns out that the inverse Jordan structure problem is always solvable in the classof truncated Toeplitz operators. In fact, we get a bit more.
Theorem 6.1.
Every operator on a finite dimensional space is similar to a co-analytic truncated Toeplitz operator.Proof.
Recalling the notation (2.18), for a finite Blaschke product Θ, we writeΘ = ϕ d z ϕ d z · · · ϕ d r z r , (6.2) where z , z , . . . , z r are the distinct zeros of Θ, and d := d + d + · · · + d r is theorder of Θ. Let Q := { A ψ ∈ T Θ : ψ ∈ H ∞ } denote the algebra of co-analytic truncated Toeplitz operators on K Θ . Note that Q is the set of A p where p is a polynomial of degree at most d .For 1 ≤ i ≤ r, let P i be the Riesz idempotent corresponding to the eigenvalue z i of A z and note that P i ∈ Q and ran P i = ker( A z − z i I ) d i [8, p. 569]. From here itis easy to see that ran P i = K ϕ dizi (6.3)and that an orthonormal basis for this subspace is { k z i ϕ j − z i : 1 ≤ j ≤ d i } . Relative to the basis above, the restriction of A ϕ zi to K ϕ dizi has a matrix whichis a d i × d i Jordan block. Thus the algebra Q i := Q|K ϕ dizi is spatially isomorphic to the algebra of d i × d i upper triangular Toeplitz matrices.Since K Θ = K ϕ d z ⊕ K ϕ d z ⊕ · · · ⊕ K ϕ drzr , is a (non-orthogonal) direct sum of vector spaces, we see from (6.3) that Q = Q ⊕ Q ⊕ · · · ⊕ Q r , is a (non-orthogonal) direct sum of algebras. It is now clear that given a Jordancanonical form, we can find a co-analytic truncated Toeplitz operator with thatform. The number of blocks in the form is the number of distinct zeros of Θ andthe size of each block determines the multiplicity of each given zero. (cid:3) The proof of Theorem 6.1 also proves the following corollary:
Corollary 6.4. If Θ is a finite Blaschke product, Q , the co-analytic truncatedoperators on K Θ , is spatially similar to Q ∗ := { A ∗ : A ∈ Q} , the analytic truncatedToeplitz operators on K Θ .Proof. Observe that for each k , Q k and ( Q k ) ∗ are spatially isomorphic. (cid:3) Theorem 5.7 asserts that for a fixed inner function Θ, T Θ contains many normaloperators. However, they are not among the analytic (or co-analytic) truncatedToeplitz operators except in trivial cases. Proposition 6.5. If Θ is inner and A ϕ ∈ T Θ is normal and not a multiple of theidentity operator, then ϕ H ∪ H .Proof. Suppose that ϕ ∈ H and A ϕ ∈ T Θ is normal. Since A ϕ = A P Θ ϕ [23,Thm. 3.1], we can assume that ϕ ∈ K Θ . Furthermore, if K = 1 − Θ(0)Θ is thereproducing kernel for K Θ at the origin, we have A K f = P Θ ( f − f Θ(0)Θ) = f, f ∈ K Θ , and so A K = I (this identity was observed in [23, p. 499]). Since A ϕ is normal ifand only if A ϕ − aI = A ϕ − aK is normal, we can set a = ϕ (0) / k K k to assumethat A ϕ is normal with ϕ ∈ K Θ and ϕ (0) = 0 . RUNCATED TOEPLITZ OPERATORS 19
This means that ϕ = zg for some g ∈ H , and, since S ∗ ϕ = ( ϕ − ϕ (0)) /z ∈ K Θ , wesee that g ∈ K Θ .To show that A ϕ cannot be normal, we will prove the inequality (cid:13)(cid:13) A ∗ ϕ K (cid:13)(cid:13) < k A ϕ K k . Observe that A ϕ K = P Θ ( ϕ − Θ(0)Θ ϕ ) = ϕ since ϕ ∈ K Θ . Now notice that A ∗ ϕ K = P Θ ( ϕ − Θ(0) ϕ Θ)= 0 − Θ(0) P Θ (( zg )Θ)= − Θ(0) P Θ ( Cg ) ( Cg = zg Θ)= − Θ(0)
Cg.
Finally note that (cid:13)(cid:13) A ∗ ϕ K (cid:13)(cid:13) = | Θ(0) | k Cg k = | Θ(0) | k g k ( C is isometric)= | Θ(0) | k zg k = | Θ(0) | k ϕ k < k ϕ k (since | Θ(0) | < k A ϕ K k . (cid:3) References
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Department of Mathematics, University of North Carolina, Chapel Hill, North Car-olina 27599
E-mail address : [email protected] Department of Mathematics, Pomona College, Claremont, California 91711
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Department of Mathematics and Computer Science, University of Richmond, Rich-mond, Virginia 23173
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