(ProQuest: ... denotes non-US-ASCII text omitted.)
Recommended by Stephen Clark
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Received 14 May 2010; Accepted 30 August 2010
1. Introduction
Let ... be the unit disk of the complex plane ... , and let dA be the Lebesgue measure on ... . The Lebesgue space of (classes of) square summable complex-valued functions is denoted by [Lagrangian (script capital L)]2 (...,dA) . The Bergman space [Lagrangian (script capital L)]a2 is the Hilbert subspace of [Lagrangian (script capital L)]2 (...,dA) consisting of analytic functions. The orthogonal complement of [Lagrangian (script capital L)]a2 in [Lagrangian (script capital L)]2 (...,dA) is denoted by ([Lagrangian (script capital L)]a2 )[perpendicular] . The Hilbert space ([Lagrangian (script capital L)]a2 )[perpendicular] is readily seen to be not a reproducing kernel Hilbert space. This is one of the major difficulties that occurs when dealing with this space. A second one is the fact that its elements have no standard common qualities such as analyticity harmonicity, while a lesser difficulty is the complicated form of the corresponding basis.
Despite the difficulties just listed, Stroethoff and Zheng in [1, 2] have adopted new techniques to investigate various properties of a class of operators acting on ([Lagrangian (script capital L)]a2 )[perpendicular] , namely, dual Toeplitz operators. A dual Toeplitz operator is defined on ([Lagrangian (script capital L)]a2 )[perpendicular] to be a multiplication (by the symbol) followed by a projection onto ([Lagrangian (script capital L)]a2 )[perpendicular] . Although dual Toeplitz operators are different from Toeplitz operators in many respects, they do share some properties with them. But surprisingly, dual Toeplitz operators on ([Lagrangian (script capital L)]a2 )[perpendicular] resemble much more Hardy space Toeplitz operators than Bergman space Toeplitz operators. Lu in [3] and Cheng and Yu in [4] considered dual Toeplitz operators in higher dimensions; while Yu and Wu in [5] considered dual Toeplitz operators in the framework of Dirichlet spaces.
The study of the numerical ranges of Hardy space Toeplitz operators goes back to Brown and Halmos [6]. Subsequent treatment was reconsidered in Halmos' book [7]. Later on, Klein [8] showed that the numerical range depends only on the spectrum of the given Hardy space Toeplitz operator. The Bergman space case was successfully considered only twenty years later by Thukral [9] in case of bounded harmonic symbols. More recently Choe and Lee [10],as well as Gu[11], treat higher-dimensional Bergman space analogs. The case of Bergman space Toeplitz operators with bounded radial symbols has been considered very recently by Wang and Wu [12]. The connection between spectral sets and numerical ranges was considered first by Schreiber [13]. Further investigations had been pursued by Hildebrandt [14] and Clark [15]. The subnormality, and particularly the quasinormality, of Hardy space Toeplitz operators has been discussed chronologically by Itô and Wong [16], Amemiya et al. [17], Abrahamse [18], Cowen and Long [19], Cowen [20, 21], Lee [22], and Yoshino [23]. The case of Bergman space Toeplitz operators has been discussed by Faour in [24] as well as by Jim Gleason in a recent preprint.
Accordingly, in this paper, we mainly investigate qualitative properties of the numerical range of a dual Toeplitz operator. We consider various classes of such operators, such as normal and quasinormal ones. We completely describe the numerical ranges of some of them and establish the main qualitative properties of the numerical ranges of others. We also shed some light on the analog of Halmos' fifth problem on the classification of subnormal Toeplitz operators.
Our paper is organized as follows: in Section 2, we exhibit some preliminary concepts needed in the sequel. Section 3 mainly concerns the description of the numerical ranges of normal dual Toeplitz operators. Section 4 contains the characterization of the numerical ranges of the more general case of nonnormal dual Toeplitz operators with harmonic symbols. In Section 5, we give heuristic proofs of some results based on the concept of lines of support of the numerical range. In Section 6, we briefly discuss the connection between spectral sets and spectra of dual Toeplitz operators. Section 7 is devoted to the quasinormality of dual Toeplitz operators. For the case of dual Toeplitz operators, we introduce and adumbrate the analog of Halmos' fifth problem on the classification of subnormal Toeplitz operators.
2. Preliminaries
Let T be a bounded operator on a Hilbert space ... . Denote its spectrum by σ(T) . The numerical range is a prototype of the spectrum, and it proves useful whenever information about T is needed. It is defined by ...B2;(T):={...T[straight phi],[straight phi]...,[straight phi]∈..., ||[straight phi]||=1} . The main involved features of the numerical range are as follows. ...B2;(T) is always bounded and convex, (Toeplitz-Hausdorff theorem). Its closure ...B2;(T)¯ contains the spectrum σ(T) . If ...B2;(T) reduces to the singleton {λ} , then T=λI . ...B2;(T) is a linear function of T , that is, ...B2;(αT+β)=α...B2;(T)+β, ∀α,β∈... ; hence we see that ω...B2;(...AE;f )=...B2;(...AE;ωf ), for ω∈... , and that λ∈...B2;(...AE;f ) implies that 0∈...B2;(...AE;f-λ ) . If ...B2;(T) is a subset of the real axis, then T must be self-adjoint. If T is normal and ...B2;(T) is closed, then the extreme points of ...B2;(T) are eigenvalues. If T is normal, the closure of ...B2;(T) is the smallest closed convex set containing the spectrum of T . For further details on the numerical ranges as well as various applications of this pioneering tool in operator theory, see [7, 25-27].
For f∈[Lagrangian (script capital L)]∞ (...), define the dual Toeplitz operator ...AE;f to be the operator on ([Lagrangian (script capital L)]a2)[perpendicular] given by [figure omitted; refer to PDF] Here ...AC;=I-P is the familiar orthogonal projection from [Lagrangian (script capital L)]2 (...,dA) onto ([Lagrangian (script capital L)]a2 )[perpendicular] and P is the orthogonal projection from [Lagrangian (script capital L)]2 (...,dA) onto the Bergman space [Lagrangian (script capital L)]a2 . Moreover, for f∈[Lagrangian (script capital L)]∞ (...) and g∈...∞ , (the algebra of bounded analytic functions) we have [figure omitted; refer to PDF] [figure omitted; refer to PDF] where Hf :u∈[Lagrangian (script capital L)]a2 ...Hf u=...AC;(fu)∈([Lagrangian (script capital L)]a2 )[perpendicular] is the Hankel operator. If E is a subset of ..., then the convex hull of E, denoted by [Hamiltonian (script capital H)](E) , is the smallest convex set containing E. Useful properties of the convex hull are well known [26, 28]. For instance, we know that in general the convex hull of an open set is open and the convex hull of a compact set is compact. The following easy property will be also used
Remark 2.1.
If A is a bounded subset of a finite dimensional normed space, then we have [Hamiltonian (script capital H)](A)¯=[Hamiltonian (script capital H)](A¯) . Indeed, observe that A⊂A¯ implies that [Hamiltonian (script capital H)](A)⊂[Hamiltonian (script capital H)](A¯) , whence [Hamiltonian (script capital H)](A)¯⊂[Hamiltonian (script capital H)](A¯)¯=[Hamiltonian (script capital H)](A¯) . Conversely, since A¯⊂[Hamiltonian (script capital H)](A)¯ , we infer that [Hamiltonian (script capital H)](A¯)⊂[Hamiltonian (script capital H)]([Hamiltonian (script capital H)](A)¯)=[Hamiltonian (script capital H)](A)¯ , as [Hamiltonian (script capital H)](A)¯ is convex.
The following main spectral properties of ...AE;f are due to Stroethoff and Zheng [1].
Theorem 2.2.
(1 ) If f is in [Lagrangian (script capital L)]∞ (...), then ...(f)⊂σe (...AE;f )⊂σ(...AE;f )⊂[Hamiltonian (script capital H)](...(f)).
(2 ) Let f be a continuous real-valued function on ... , then σ(...AE;f )=σe (...AE;f )=f(...)¯ .
(3 ) If f is a bounded analytic or coanalytic function on ... , then σ(...AE;f )=σe (...AE;f )=f(...)¯.
For our purpose, we now prove the following useful facts about dual Toeplitz operators.
Lemma 2.3.
Let f be in [Lagrangian (script capital L)]∞ (...) . Then, ...AE;f is self-adjoint if and only if f is real.
Proof.
...AE;f is self-adjoint means that ...AE;f =...AE;f* . This is equivalent to the fact that f=f¯ , since ||...AE;g ||=||g||∞ , for g∈[Lagrangian (script capital L)]∞ (...) , which is equivalent to the fact that f is real-valued.
Lemma 2.4.
Let f be in [Lagrangian (script capital L)]∞ (...) . Then, ...AE;f ≥0 if and only if f≥0 .
Proof.
If f≥0 , then ......AE;f g,g...=...Q(fg),g...=...fg,g...=∫... f(z)|g(z)|2 dA(z)≥0,∀g∈([Lagrangian (script capital L)]a2 )[perpendicular] . Conversely, suppose that ...AE;f ≥0 , then in particular its spectrum lies in [0,∞) . By part (1) of Theorem 2.2, we obtain ...(f)⊂σ(...AE;f )⊂[0,∞) , whence f≥0 .
Corollary 2.5.
(i ) A bounded dual Toeplitz operator with a real spectrum must be self-adjoint.
(ii ) A bounded dual Toeplitz operator with spectrum lying in the positive real half-axis must be positive.
Proof.
This result follows from part (1) of Theorem 2.2 and Lemmas 2.3 and 2.4.
Theorem 2.6.
Let f be a nonconstant bounded harmonic real-valued function on ... . Then, the operator ...AE;f has no eigenvalues.
Proof.
Since in general for a constant λ we have ...AE;f -λ=...AE;f-λ and f-λ is harmonic if f is, it suffices to show that ...AE;f g=0 implies that g=0, ∀g∈([Lagrangian (script capital L)]a2 )[perpendicular] . If ...AE;f g=...AC;(fg)=0 , then fg∈[Lagrangian (script capital L)]a2 . Let h∈...∞ (...) , then h¯g∈([Lagrangian (script capital L)]a2 )[perpendicular] . Indeed, ||h¯g||2 ≤||h||∞ ||g||2 , and ...h¯g,u...=...g,hu...=0, ∀u∈[Lagrangian (script capital L)]a2 , (as hu∈[Lagrangian (script capital L)]a2 and g∈([Lagrangian (script capital L)]a2 )[perpendicular] ). It follows that ...fg,h¯g...=0 . Taking real parts and noticing that f is real, we see that [figure omitted; refer to PDF] Since ...e(...∞ ) is weak * -dense in the set of bounded real harmonic functions [29], we can replace ...e(h) with f in the latter to obtain f2 |g|2 =0 . This implies that f2 and |g|2 have disjoint supports. However, the harmonic function f cannot vanish on a set of positive measure without being zero, whence g=0 .
3. Characterization of the Numerical Range
An operator T:M[arrow right]M is said to be subnormal if it admits an extension S:E[arrow right]E , such that M⊂E , S is normal, and M is invariant under S . It is well known that if an operator T is subnormal, then it is convexoid, that is, ...B2;(T)¯=[Hamiltonian (script capital H)](σ(T)) . First, the following observation is worth stressing.
Proposition 3.1.
Suppose that f∈...∞ (...) . Then ...AE;f and ...AE;f¯ are convexoid, that is, ...B2;(...AE;f )¯=[Hamiltonian (script capital H)](σ(...AE;f )) , and ...B2;(...AE;f¯ )¯=[Hamiltonian (script capital H)](σ(...AE;f¯ )) .
Proof.
Let f∈...∞ (...) , then the multiplication operator [physics M-matrix]f¯ :[Lagrangian (script capital L)]2 (...)[arrow right][Lagrangian (script capital L)]2 (...) is a normal extension of ...AE;f¯ . Indeed, we have [physics M-matrix]f¯[physics M-matrix]f =[physics M-matrix]f[physics M-matrix]f¯ and [physics M-matrix]f¯ (([Lagrangian (script capital L)]a2)[perpendicular] )={f¯g,g∈([Lagrangian (script capital L)]a2)[perpendicular] }⊂([Lagrangian (script capital L)]a2)[perpendicular] . Moreover, it is clear that ...AE;f¯ is the restriction of [physics M-matrix]f¯ to ([Lagrangian (script capital L)]a2)[perpendicular] . Thus ...AE;f¯ is subnormal, whence ...B2;(...AE;f¯ )¯=[Hamiltonian (script capital H)](σ(...AE;f¯ )) .
Concerning the other part of the Proposition, of course ...AE;f is not necessarily subnormal, nevertheless we obtain a similar result by exploring the fact that ...B2;(...AE;f )={λ¯,λ∈...B2;(...AE;f¯ )} and σ(...AE;f )={λ¯,λ∈σ(...AE;f¯ )}. Hence, we obtain ...B2;(...AE;f )¯=[Hamiltonian (script capital H)](σ(...AE;f )) .
For bounded analytic or coanalytic symbols, the fact that ...AE;f is convexoid comes from the subnormality of this operator as Proposition 3.1 asserts. However, the spectral inclusion theorem, (namely, part (1) of Theorem 2.2), refines this result. Indeed, making use of the spectral inclusion property, it turns out that all bounded dual Toeplitz operators are convexoid; this represents the aim of the following assertion.
Proposition 3.2.
The closure of the numerical range of a bounded dual Toeplitz operator is the convex hull of its spectrum, that is, ...B2;(...AE;f )¯=[Hamiltonian (script capital H)](σ(...AE;f )) , for f∈[Lagrangian (script capital L)]∞ (...) .
Proof.
Consider the multiplication operator [physics M-matrix]f on [Lagrangian (script capital L)]2 (...,dA) . It is known to be normal, whence convexoid. Thus ...B2;([physics M-matrix]f )¯=[Hamiltonian (script capital H)](σ([physics M-matrix]f )) . By Problem 67 of [7], we see that σ([physics M-matrix]f )=...(f) . By the spectral inclusion Theorem, we see that σ([physics M-matrix]f )=...(f)⊂σ(...AE;f ) . This yields the following inclusions ...B2;([physics M-matrix]f )¯=[Hamiltonian (script capital H)](σ([physics M-matrix]f ))=[Hamiltonian (script capital H)](...(f))⊂[Hamiltonian (script capital H)](σ(...AE;f )) . Now, since [physics M-matrix]f is the minimal normal dilation of ...AE;f , we see that ...B2;(...AE;f )¯⊂...B2;([physics M-matrix]f )¯ , which is clear from the definition of the numerical range and the fact that ...AE;f is the compression of [physics M-matrix]f . Therefore, we obtain the first inclusion ...B2;(...AE;f )¯⊂[Hamiltonian (script capital H)](σ(...AE;f )) .
The reverse inclusion is easier: we have σ(...AE;f )⊂...B2;(...AE;f )¯ , which implies that [Hamiltonian (script capital H)](σ(...AE;f ))⊂[Hamiltonian (script capital H)](...B2;(...AE;f )¯)=...B2;(...AE;f )¯ , since ...B2;(...AE;f )¯ is convex.
Remark 3.3.
In connection with Proposition 3.2, we ask whether all elements of the dual Toeplitz algebra ...9F;...AF; , (the C*-algebra generated by {...AE;f ,f∈...∞ } ), are convexoid, according to the fact that it is generated by subnormal operators.
Now, we are going to characterize the numerical ranges of dual Toeplitz operators with bounded harmonic symbols. First, we make the following observation.
Remark 3.4.
According to part (2) of Theorem 2.2, for a nonconstant bounded continuous real-valued function f on ... , we infer that σ(...AE;f )=...(f)=f(...)¯ is an interval. As f is bounded, we deduce that σ(...AE;f )=[inf f,sup f] . Obviously, we have [Hamiltonian (script capital H)](σ(...AE;f ))=[inf f,sup f] .
Lemma 3.5.
Suppose that f is a nonconstant bounded harmonic real-valued function on ... . Then one has ...B2;(...AE;f )=(inf f,sup f) .
Proof.
By Proposition 3.2 and Remark 3.4, we see that ...B2;(...AE;f )¯=[inf f,sup f] , (the continuity is redundant in the harmonicity). Since ...B2;(...AE;f ) is a convex set whose closure is [inf f,sup f] , ...B2;(...AE;f ) contains all elements of [inf f,sup f] except possibly the extreme points inf f and sup f . Thus (inf f,sup f)⊂...B2;(...AE;f ) .
Now, suppose that either inf f or sup f belongs to ...B2;(...AE;f ) . Then it is an extreme point of ...B2;(...AE;f ) , which in fact must be an eigenvalue of ...AE;f . However, Theorem 2.6 tells us that such ...AE;f has no eigenvalues. Thus, we should have ...B2;(...AE;f )=(inf f,sup f) .
Parallel to Brown and Halmos [6] characterization of normal Hardy space Toeplitz operators as well as Axler and Cuckovic [30] one of normal Bergman space Toeplitz operators with bounded harmonic symbols, normal dual Toeplitz operators were characterized in [1] as follows: for a bounded measurable function f on ... , the dual Toeplitz operator ...AE;f is normal if and only if the range of f lies on a line. Accordingly, we are able to characterize the numerical range of normal dual Toeplitz operators with bounded harmonic symbols.
Theorem 3.6.
Let f be a nonconstant bounded harmonic function on ... , and suppose that ...AE;f is normal. Then, there are (complex) numbers a,b such that σ(...AE;f ) is the closed line segment [a,b] and ...B2;(...AE;f ) is the corresponding open line segment (a,b) .
Proof.
Taking into account the assumption that ...AE;f is normal, we are certain from the existence of (complex) constants α,β and a real-valued function g such that f=αg+β whence ...AE;f =α...AE;g +β . From the harmonicity of f and the linearity of the Laplacian, we see that g must be bounded harmonic and real-valued. Now, Lemma 3.5 asserts, therefore, that σ(...AE;g )=[m,M] and ...B2;(...AE;g )=(m,M) , where m=inf g and M=sup g . Thus σ(...AE;f )=[a,b] and ...B2;(...AE;f )=(a,b) , (line segments in ... ), with a=αm+β , and b=αM+β .
4. The Numerical Range of a Nonnormal Dual Toeplitz Operator
Lemma 2.4 has a nice consequence on the self-adjointness of certain dual Toeplitz operators.
Theorem 4.1.
Let f∈[Lagrangian (script capital L)]∞ (...) be harmonic. Suppose that ...B2;(...AE;f ) lies in the complex upper half-plane and contains some real number. Then ...AE;f must be self-adjoint.
Proof.
The numerical range lies in the upper half-plane means that ...m(......AE;f g,g...)≥0 , for ||g||=1 . Taking g/||g||2 if necessary, we may conclude that ...m(......AE;f g,g...)≥0,∀g∈([Lagrangian (script capital L)]a2 )[perpendicular] . Since ......AE;f g,g...=∫... f|g|2 dA,∀g∈([Lagrangian (script capital L)]a2 )[perpendicular] , taking imaginary parts, we obtain [figure omitted; refer to PDF] This happens only if ...m(f)≥0 , by Lemma 2.4. Now, suppose that a real number c∈...B2;(...AE;f ) , then there exists some h∈([Lagrangian (script capital L)]a2 )[perpendicular] , with ||h||=1 , such that ......AE;f h,h...=c . Writing c in the form c=c...h,h... , we obtain ∫... (f-c)hh¯dA=0. Again, taking imaginary parts we obtain [figure omitted; refer to PDF] As ...m(f)≥0 , we deduce that ...m(f)|h|2 =0 on ... . This implies that ...m(f) and |h|2 have disjoint supports. Since ||h||=1 , we deduce that h≠0 on ... . Thus supp (...m(f)) has a positive measure, that is, the harmonic function ...m(f) must be zero on a set of nonzero measure. It follows that ...m(f)=0 on ... , whence ...AE;f is self-adjoint by Lemma 2.3.
Regarding the numerical ranges of a certain class of dual Toeplitz operators, we have the following qualitative characterization.
Theorem 4.2.
Let f∈[Lagrangian (script capital L)]∞ (...) be harmonic such that ...AE;f is nonnormal. Then ...B2;(...AE;f ) is an open convex set.
Proof.
We need only to verify that the convex set ...B2;(...AE;f ) is open. To see this, we proceed by contradiction and suppose that it is not open. Hence it intersects its boundary ∂...B2;(...AE;f ) . Let λ be one of such points, that is, λ∈...B2;(...AE;f )∩∂...B2;(...AE;f ) , which can be rewritten as 0∈...B2;(...AE;f-λ )∩∂...B2;(...AE;f-λ ) . Now, the convexity of ...B2;(...AE;f-λ ) and the fact that 0∈∂...B2;(...AE;f-λ ) enable us to rotate it so that it lies in the upper half-plane. This means that there exists a unimodular complex number ω=eiθ , for some θ∈[0,2π] , such that ω...B2;(...AE;f-λ )=...B2;(...AE;ωf-ωλ ) lies in the upper half-plane. By Theorem 4.1, ...AE;ωf-ωλ must be self-adjoint. In other words there exists a real-valued function g on ... such that ωf-ωλ=g . This implies that f=αg+β , for some constants α and β . So, we infer that ...AE;f =α...AE;g +β is normal, which contradicts the original hypothesis and completes the proof.
Remark 4.3.
Note that Theorems 4.1 and 4.2, (as well as their subsequent corollaries), remain still valid if one assumes merely that the symbols have harmonic imaginary parts.
Corollary 4.4.
If f is a bounded analytic or coanalytic function, then ...B2;(...AE;f )=[Hamiltonian (script capital H)](f(...)).
Proof.
If f is constant, the fact is trivially satisfied. If f is not constant, then ...AE;f is not normal (because the range of an analytic or coanalytic function cannot lie on a line). Hence, by Theorem 4.2, ...B2;(...AE;f ) is an open convex set. On the other hand, by the open mapping theorem f(...) is open whence [Hamiltonian (script capital H)](f(...)) is an open convex set too. Now, by Proposition 3.1 and part (3) of Theorem 2.2 as well as Remark 2.1, we see that ...B2;(...AE;f )¯=[Hamiltonian (script capital H)](f(...)¯)=[Hamiltonian (script capital H)](f(...))¯. Since an open convex planar set is the interior of its closure, we infer that ...B2;(...AE;f ) coincides with the convex hull of f(...) .
5. Aesthetic Consequences Using Lines of Support of ...B2;(...AE;f )
A line L is said to be a line of support of a planar convex set K at a boundary point P∈∂K , if P∈L and the set K is contained in the closure of one of the two half-planes into which L cuts the plane. Clearly, every point on the boundary of a planar convex set lies on a line of support. Based on the concept of lines of support of the numerical range ...B2;(...AE;f ) , several interesting consequences of Theorem 4.1 can be observed. Note that the underlying idea goes back to Brown and Halmos [6].
Corollary 5.1.
Let f∈[Lagrangian (script capital L)]∞ (...) be harmonic. If a line of support of the numerical range ...B2;(...AE;f ) of the dual Toeplitz operator ...AE;f contains a point of ...B2;(...AE;f ) , then it contains its whole spectrum σ(...AE;f ) (and hence its entire numerical range ...B2;(...AE;f ) ).
Corollary 5.2.
Let f∈[Lagrangian (script capital L)]∞ (...) be harmonic. If the spectrum σ(...AE;f ) consists of merely a finite number of eigenvalues, then the operator ...AE;f is scalar.
Proofs
Proof of Corollary 5.1
With regard to Corollary 5.1, clearly the line of support can be rotated along with the numerical range till the line will be horizontal and the numerical range above it. Then we translate both in such a way that the line of support will be the real axis; (these two operations correspond to a linear function of the form z[arrow right]eiθ z+ω , for a fixed complex number ω and a fixed θ∈[0,2π] ). Then one can apply Theorem 4.1 to conclude that eiθ...AE;f +ω is self-adjoint, whence eiθ σ(...AE;f )+ω⊂... , and therefore σ(...AE;f ) lies on the original line of support. Since ...B2;(...AE;f )⊂...B2;(...AE;f )¯=[Hamiltonian (script capital H)](σ(...AE;f )) , by Proposition 3.2, we infer that ...B2;(...AE;f ) lies on the relevant line of support.
Proof of Corollary 5.2
Concerning Corollary 5.2, if σ(...AE;f ) consists of a finite number of eigenvalues, then [Hamiltonian (script capital H)](σ(...AE;f )) is a polygon with vertices of some of such values. Thus, one can find at least a line of support passing through one of such vertices, which must be in fact an extreme point of ...B2;(...AE;f ) . Hence, making use of Corollary 5.1, we infer that all those eigenvalues of ...AE;f lie on such line. Consequently, the numerical range is a segment on this line with two extreme points. Taking the line of support perpendicular to the above line, and passing by an endpoint of the segment, and repeating the same procedure, we infer that the numerical range lies on the new line of support as well. A segment lying in two perpendicular lines must be a single point. So, σ(...AE;f ) must be a singleton. By Proposition 3.2 and the properties of the numerical range, we infer that ...AE;f must be scalar.
Remark 5.3.
Suppose that ...AE;f is compact. We know that its spectrum consists of at most a countable number of eigenvalues, including 0 . Arguing in the same manner as in the proof of Corollary 5.2, we can show that there are no nonzero compact dual Toeplitz operators with bounded harmonic symbols, which is a particular case of [1, Theorem 7.5 ].
6. Connection with Spectral Sets
A compact subset E⊂... containing the spectrum σ(T) of a bounded linear operator T acting on a given Hilbert space is called a k -spectral set for T if [figure omitted; refer to PDF] holds for every function ...(z) analytic in a neighborhood of E (in [13], rational functions are used instead), where the first norm in the above inequality is the operator norm and the second one is the sup norm over E . In particular, if k=1 , 1 -spectral sets are simply called spectral sets. For more details on spectral sets, we refer to [13-15, 31] and the references therein. For instance, the spectrum of any subnormal operator is a spectral set. The closed unit disk is a spectral set for any contraction.
Our main concern in this section is to describe the dual Toeplitz operators analog of an interesting connection, pointed out by Schreiber [13], between spectral sets and numerical ranges of Toeplitz operators. Let us start with some trivial situations, where σ(...AE;[straight phi] ) is spectral
(1) If ...AE;[straight phi] is a bounded normal dual Toeplitz operator, (i.e., [straight phi](...) lies on a line in ... [1]), then σ(...AE;[straight phi] ) is a spectral set.
(2) If [straight phi] is a bounded coanalytic function on the unit disk ... , then σ(...AE;[straight phi] ) is a spectral set, (since it is subnormal by the proof of Proposition 3.1).
(3) If the spectrum of a bounded dual Toeplitz operator ...AE;[straight phi] is a disk, then σ(...AE;[straight phi] ) is a spectral set too.
However, we can observe that the spectra of dual Toeplitz operators with analytic symbols are also spectral sets for their corresponding operators. This follows from the following observation. If E⊂... is a planar subset, set E* ={ω¯,ω∈E} , in particular it can be easily verified that σ(T* )=σ(T)* . Also we adopt the notation f* (z)=f(z¯)¯ which is an analytic function on E* whenever f is analytic on E . Then we have the following.
Lemma 6.1.
Let T be a bounded linear operator on a given Hilbert space. Then, σ(T) is a k -spectral set for T if and only if σ(T)* is a k -spectral set for its adjoint T* .
Proof.
The fact that σ(T) is a k -spectral set of T means that Inequality (6.1) holds, for any ... holomorphic in a neighborhood of σ(T) . Notice that from the definition of the holomorphic functional calculus we have [figure omitted; refer to PDF] On the other hand, we have [figure omitted; refer to PDF] Combining Inequality (6.1) and the last two identities, we infer that [figure omitted; refer to PDF] In order to establish the equivalence, it suffices to observe that if ... is holomorphic in a neighborhood of σ(T) , then ...* is holomorphic in the conjugate of the same neighborhood which contains σ(T)* and conversely.
Therefore, from the above discussion, we deduce the following.
Corollary 6.2.
If [straight phi] is a bounded analytic function on the unit disk ... , then σ(...AE;[straight phi] ) is a spectral set for ...AE;[straight phi] .
Similar results for coanalytic Toeplitz operators on both Hardy and Bergman spaces can also be inferred.
Corollary 6.3.
(i ) If [straight phi]¯∈...∞ (∂...) , then σ(T[straight phi] ) is a spectral set for T[straight phi] defined on ...2 (∂...) .
(ii ) If [straight phi] is a bounded coanalytic function on the unit disk ... , then σ(T[straight phi] ) is a spectral set for T[straight phi] defined on [Lagrangian (script capital L)]a2 (...) .
7. Some Thoughts on Quasinormal Dual Toeplitz Operators
The Bergman space [Lagrangian (script capital L)]a2 has normalized reproducing kernel kw given by [figure omitted; refer to PDF] Recall that for w∈..., the involutive disk automorphism [straight phi]w is defined by [figure omitted; refer to PDF] For a linear operator T on ([Lagrangian (script capital L)]a2 )[perpendicular] and w∈..., define the operator ...w (T)...=T-...AE;[straight phi]w TS[straight phi]¯w . A second application of it gives [figure omitted; refer to PDF] For f,g∈[Lagrangian (script capital L)]2 (...,dA), define the rank one operator (f[ecedil]7;g) by [figure omitted; refer to PDF]
If T1 and T2 are bounded linear operators on [Lagrangian (script capital L)]2 (...,dA), then for f , g ∈[Lagrangian (script capital L)]2 (...,dA), we have [figure omitted; refer to PDF] In the sequel, we will need a formula relating the image of the product HfHg¯* under the action of the operator ...w2 and the functions Hf (kw ) and Hg¯ (kw ) . For f,g∈[Lagrangian (script capital L)]∞ (...), combining (2.3), (7.3), and (7.5), (for a detailed proof see [1, 2]), we obtain [figure omitted; refer to PDF] An operator T on a Hilbert space is called quasinormal if it commutes with T* T . It is well known that quasinormal operators are subnormal. In what follows we are going to show that there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols that are not normal.
Theorem 7.1.
Let f be in ...∞ (...) , and suppose that ...AE;f is quasinormal. Then, the symbol f must be constant.
Proof.
If f=0 , the conclusion is obvious. So, suppose that f≠0 and that ...AE;f is quasinormal, then we have [figure omitted; refer to PDF] Since f is analytic, using Relations (2.2), we obtain [figure omitted; refer to PDF] [figure omitted; refer to PDF] Now, (7.7)-(7.9) yield [figure omitted; refer to PDF] Introducing the operator ...ω , by (7.6), we see that [figure omitted; refer to PDF] Combining the latter with (7.10), we obtain H|f|2kω [ecedil]7;Hf¯kω =Hf¯kω [ecedil]7;Hf¯2kω . Since k0 =1 , taking ω=0 , we obtain H|f|2 1[ecedil]7;Hf¯ 1=Hf¯ 1[ecedil]7;Hf¯2 1. In other words, by the definition of the rank one operator, we have [figure omitted; refer to PDF] If Hf¯ 1=0 , then f¯∈[Lagrangian (script capital L)]a2 . Thus f¯ must be constant, whence the result follows immediately. If Hf¯ 1≠0 , we distinguish several cases as follows.
(i) The case H|f|2 1=0 and Hf¯2 1=0 cannot happen (because Hf¯ 1=...AC;(f¯)=f¯-f¯(0) and Hf¯2 1=...AC;(f¯2 )=f¯2 -f¯2 (0) vanish simultaneously).
(ii) The similar case H|f|2 1≠0 and Hf¯2 1=0 is impossible too, for the same reason.
(iii): The case H|f|2 1=0 and Hf¯2 1≠0 is impossible too, otherwise ...u,Hf¯2 1...=0, for all u∈([Lagrangian (script capital L)]a2 )[perpendicular] ; thus Hf¯2 1∈[Lagrangian (script capital L)]a2 too, whence Hf¯2 1=0 contradicting the assumption.
(iv) If H|f|2 1≠0 and Hf¯2 1≠0 , then clearly from (7.12) there exists some nonzero constant λ∈... , such that
[figure omitted; refer to PDF] Rephrasing (7.13), we see that ...AC;(f¯-λ|f|2 )=0 and ...AC;(f¯-λ¯ f¯2 )=0 . Thus, f¯-λ|f|2 and f¯-λ¯ f¯2 are in [Lagrangian (script capital L)]a2 , (they are analytic in particular). But, since f is analytic, we see that f(1-λf)=f-λf2 is analytic too, whence it is constant. So, set f(1-λf)=μ , for some nonzero complex constant μ . On the other hand, since f¯-λ|f|2 =f¯(1-λf) is analytic, multiplying by the analytic function f2 , we obtain an analytic function, namely, f¯(1-λf)f2 =|f|2 (1-λf)f=μ|f|2 . Now, the function μ|f|2 , (whose range lies on a line, as |f|2 is real-valued and μ is constant), can be analytic only if it is constant; whence we infer that |f|2 is constant. Finally, it is well known that an analytic function with a constant modulus must be constant, whence f must be constant.
For bounded conjugate analytic symbols the matter is much more simpler and it uses the Brown-Halmos type Theorem (namely, [1, Theorem 3.1 ]). Indeed, we have the following.
Theorem 7.2.
Let f be in ...∞ (...)¯ , and suppose that ...AE;f is quasinormal. Then, f must be constant.
Proof.
If f=0 , the matter is obvious; so suppose that f≠0 . Since f is coanalytic, using Relations (2.2) and (2.3), we obtain [figure omitted; refer to PDF] [figure omitted; refer to PDF] Suppose that ...AE;f is quasinormal, then from (7.7), (7.14), and (7.15), we see that ...AE;f...AE;|f|2 =...AE;f|f|2 . Hence ...AE;f...AE;|f|2 is a dual Toeplitz operator. By the Brown-Halmos type Theorem ([1, Theorem 3.1 ]), we infer that either f is analytic or |f|2 is coanalytic. If f is analytic, then it is constant since it is coanalytic by hypothesis. If the real function |f|2 is coanalytic, then it is constant; whence f is constant as well, (as a coanalytic function with constant modulus). Thus, in all cases, we infer that f is constant.
Remark 7.3.
Clearly if f is constant, then ...AE;f is normal and then it is quasinormal. So, Theorems 7.1 and 7.2 can be expressed as follows. Let f be a bounded analytic, (or coanalytic), function. Then, ...AE;f is quasinormal if and only if f is constant, that is, there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols that are not normal.
Corollary 7.4.
Let f be a bounded analytic or coanalytic function, and suppose that ...AE;f is quasinormal. Then, the numerical range of ...AE;f reduces to a singleton, that is, ...B2;(...AE;f )={λ} for some complex constant λ .
Proof.
Just observe that from Proposition 3.1 one has ...B2;(...AE;f )¯=[Hamiltonian (script capital H)](σ(...AE;f )) . If ...AE;f is quasinormal, by Theorems 7.1 and 7.2, we infer that f=λ for some complex constant λ . The convex hull of a singleton is the set itself. Hence, we obtain ...B2;(...AE;f )¯=[Hamiltonian (script capital H)]({λ})={λ} .
It seems to be of interest to consider also the typical case of bounded harmonic symbols. It is well known that such functions can be decomposed as f=f1 +f2 ¯ , with Bloch functions f1 and f2 . Here, we confine ourselves to the related case, where f=g+λg¯ , for g∈...∞ (...) and λ a complex constant.
Proposition 7.5.
Suppose that f=g+λg¯, 0≠λ∈... , and 0≠g∈...∞ (...) . If ...AE;f is quasinormal, then ...AE;f is normal and λ must be unimodular.
Proof.
Since ...AE;f is quasinormal, (7.7) holds. Taking adjoints, we obtain [figure omitted; refer to PDF] Therefore, using the Hilbert space orthogonality relations, we see that ker ([...AE;f¯ ,...AE;f ])=...an(...AE;f )¯ , which must be nontrivial. Next, a couple of manipulations lead to [figure omitted; refer to PDF] Now, (2.3) as well as the fact that g is analytic yields [...AE;g¯ ,...AE;g ]=-Hg¯Hg¯* , whence [figure omitted; refer to PDF] Since the Hankel operator Hg¯ is one-to-one (as g∈...∞ ), we infer that Hg¯Hg¯* has a trivial kernel. This contradicts the fact that [...AE;f¯ ,...AE;f ] has a nontrivial kernel, unless [...AE;f¯ ,...AE;f ]=0 , which happens only if |λ|2 =1 . Thus ...AE;f is normal and λ must be unimodular.
Corollary 7.6.
Let f be as in Proposition 7.5 and suppose that ...AE;f is quasinormal. Then, ...B2;(...AE;f )=(a,b) for some complex constants a and b .
Proof.
Combining Proposition 7.5 and Theorem 3.6, the result follows.
At this stage, we would like to conclude with a crucial point, which probably sheds some light on the fifth Halmos' problem [32]. This problem asks whether every subnormal Hardy space Toeplitz operator is either normal or analytic. In the Hardy space setting, the original general problem was weakened to whether every quasinormal Toeplitz operator is either normal or analytic, and it was completely solved positively by Amemiya et al. [17], whereas Cowen and Long [19] answered the original problem in the negative. For further results in this direction, see [16, 18, 20-23]. The Bergman space analog seems to be still pending [24]. However, for dual Toeplitz operators, a similar conjecture can be stated with slight modifications, namely: every quasinormal dual Toeplitz operator must be normal. Theorems 7.1 and 7.2, Proposition 7.5 as well as Remark 7.3 support this conjecture.
Acknowledgments
The author would like to express his sincere thanks to the referees for their valuable comments and suggestions. This paper was supported by College of Science-Research Center Project no. (Math/2007/27).
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Abstract
The study of dual Toeplitz operators was elaborated by Stroethoff and Zheng (2002), where various corresponding algebraic and spectral properties were established. In this paper, we characterize numerical ranges of certain classes of dual Toeplitz operators. Moreover, we introduce the analog of Halmos' fifth classification problem for quasinormal dual Toeplitz operators. In particular, we show that there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols which are not normal.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer