解析Toeplitz代数的本质换位及其相关问题

在本文中，我们决定出多复变Ｈａｒｄｙ空间Ｈ２上解析Ｔｏｅｐｌｉｔｚ代数的本质换位．即一个算子与所有解析Ｔｏｅｐｌｉｔｚ算子本质可换，当且仅当它是符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的紧扰动．由此，符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（Ａｃ）在Ｃａｌｋｉｎ代数中的像是极大可换闭代数，这导致了Ｌ．Ｃｏｂｕｒｎ正合列的极大扩充．从这个事实，证明了符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的本质谱是连通的，这大大改进了Ｃ－Ｓ最近的工作．从本文的主要定理，证明了Ｔｏｅｐｌｉｔｚ代数Ｆ（Ｌ∞）的本质换位和本质中心是由符号属于ＱＣ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（ＱＣ），这些结果又导致了对代数Ｆ（Ｈ∞）＋Ｋ自同构群的确定．     

Toeplitz operators on Bergman spaces

Let ? be an element in \(H^\infty (D) + C(\overline D )\) such that ?^* is locally sectorial. In this paper it is shown that the Toeplitz operator defined on the Bergman spaceA ² (D) is Fredholm. Also, it is proved that ifS is an operator onA ²(D) which commutes with the Toeplitz operatorT _? whose symbol ? is a finite Blaschke product, thenS H ^∞ (D) is contained inH ^∞ (D). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman spaceA ² (D), is not connected.     

Commutants of nest algebras modulo the compact operators

Any operatorx which commutes modulo the compact operators with a nest algebra is of the form λI+C, where λ is a scalar andC is a compact operator. Any derivation from a nest algebra on a Hilbert spaceH into the compact operators onH is implemented by a compact operator. Any derivation on a quasitriangular operator algebra is inner.     

Toeplitz operators in several complex variables

Let S be the unit sphere in Cⁿ. We investigate the properties of Toeplitz operators on S, i.e., operators of the form Tφf = P(φf) where φ?L^∞(S) and P denotes the projection of L²(S) onto H²(S). The aim of this paper is to determine how far the extensive one-variable theory remains valid in higher dimensions. We establish the spectral inclusion theorem, that the spectrum of T_φ contains the essential range of φ, and obtain a characterization of the Toeplitz operators among operators on H²(S) by an operator equation. Particular attention is paid to the case where φ ? H^∞(S) + C(S) where C(S) denotes the algebra of continuous functions on S. Finally we describe a class of Toeplitz operators useful for providing counterexamples—in particular, Widom's theorem on the connectedness of the spectrum fails when n > 1.     

Cellular-indecomposable subnormal operators

A bounded operator T is cellular-indecomposable if LnM{0} whenever L and M are any two nonzero invariant subspaces for T. We show that any such subnormal operator has a cyclic normal extension and is unitarily equivalent modulo the compact operators to an analytic Toeplitz operator whose symbol is a weak-star generator of H.Dedicated to the memory of James P. WilliamsThis work was supported in part by a grant from the National Science Foundation.     

Commutants of nest algebras modulo the compact operators

Any operatorx which commutes modulo the compact operators with a nest algebra is of the form λI+C, where λ is a scalar andC is a compact operator. Any derivation from a nest algebra on a Hilbert spaceH into the compact operators onH is implemented by a compact operator. Any derivation on a quasitriangular operator algebra is inner.     

Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for SφSφψ= Sφψ.     

On composition operators onH p -spaces in several variables

In this paper, we prove that a composition operator onH ^p (B) is Fredholm if and only if it is invertible if and only if its symbol is an automorphism onB, and give the representation of the spectra of a class of composition operators. In addition, using composition operator, we discuss intertwining Toeplitz operators. Supported by NNSF and PDSF     

Spectral theory for algebraic combinations of Toeplitz and composition operators

We determine the essential spectra of algebraic combinations of Toeplitz operators with continuous symbol and composition operators induced by a class of linear-fractional non-automorphisms of the unit disk. The operators in question act on the Hardy space H² on the unit disk. Our method is to realize the C^*-algebra that they generate as an extension of the compact operators by a concrete C^*-algebra whose invertible elements are easily characterized.     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω?C, and in Fock spacesA ²(C ^N),N≧1.     

Bounded symbols and Reproducing Kernel Thesis for truncated Toeplitz operators

Compressions of Toeplitz operators to coinvariant subspaces of H² are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the Reproducing Kernel Thesis. We show that in general the answer to the first question is negative, and we exhibit some classes of spaces for which the answers to both questions are positive.     

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

Linear fractional composition operators on H2

If ? is an analytic function mapping the unit diskD into itself, the composition operatorC _? is the operator onH ² given byC _?f=fo?. The structure of the composition operatorC _? is usually complex, even if the function ? is fairly simple. In this paper, we consider composition operators whose symbol ? is a linear fractional transformation mapping the disk into itself. That is, we will assume throughout that $$\varphi \left( z \right) = \frac{{az + b}}{{cz + d}}$$ for some complex numbersa, b, c, d such that ? maps the unit diskD into itself. For this restricted class of examples, we address some of the basic questions of interest to operator theorists, including the computation of the adjoint.     

Products of composition and analytic Toeplitz operators

We will consider the problem of which the products of composition and analytic Toeplitz operators would be bounded or compact on the Hardy space H² and the Bergman space L_a².     

Every composition operator is (mean) asymptotically Toeplitz

Nazarov and Shapiro recently showed that, while composition operators on the Hardy space H² can only trivially be Toeplitz, or even “Toeplitz plus compact,” it is an interesting problem to determine which of them can be “asymptotically Toeplitz.” I show here that if “asymptotically” is interpreted in, for example, the Cesàro (C,α) sense (α>0), then every composition operator on H² becomes asymptotically Toeplitz.     

Cellular-indecomposable subnormal operators II

This work continues that begun in [9]. Our investigation has led us to the following conjecture: a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator of H. In this paper some particular cases of the conjecture are verified.This work was supported in part by a grant from the National Science Foundation.     

Unitary equivalence of one-parameter groups of Toeplitz and composition operators

The composition operators on H² whose symbols are hyperbolic automorphisms of the unit disk fixing ±1 comprise a one-parameter group and the analytic Toeplitz operators coming from covering maps of annuli centered at the origin whose radii are reciprocals also form a one-parameter group. Using the eigenvectors of the composition operators and of the adjoints of the Toeplitz operators, a direct unitary equivalence is found between the restrictions to zH² of the group of Toeplitz operators and the group of adjoints of these composition operators. On the other hand, it is shown that there is not a unitary equivalence of the groups of Toeplitz operators and the adjoints of the composition operators on the whole of H², but there is a similarity between them.     

Bifurcation for variational problems when the linearisation has no eigenvalues

For a bounded analytic function, ?, on the unit disk, $\text{D,}\text{let}\text{T}{}_{\mathrm{?}}\text{and}\text{M}{}_{\mathrm{?}}$ denote the operators of multiplication by ? on H²(?D) and L²(?D), respectively. In their 1973 paper, Deddens and Wong asked whether there is an analytic Toeplitz operator $\text{T}{}_{\mathrm{?}}$ that commutes with a nonzero compact operator, and whether every operator that commutes with an analytic Toeplitz operator has an extension that commutes with the corresponding multiplication operator on L². In the first part of this paper, we give an explicit example of an analytic Toeplitz operator T_φ that settles both of these questions. This operator commutes with a nonzero compact operator (a composition operator followed by an analytic Toeplitz operator). The only operators in the commutant of T_φ that extend to commute with M_φ are analytic Toeplitz operators. Although the commutant of T_φ contains more than just analytic Toeplitz operators, T_φ is irreducible. The remainder of the paper seeks to explain more fully the phenomena incorporated in this example by introducing a class of analytic functions, including the function φ, and giving additional conditions on functions g in the class to determine whether T_g commutes with nonzero compact operators, whether T_g is irreducible, and which operators in the commutant of T_g extend to the commutant of M_g. In particular, we find representations for operators in the commutant and second commutant of T_g.     

Invertibility of Toeplitz operators and corona conditions in a strip

A Toeplitz operator with symbol G such that detG=1 is invertible if there is a non-trivial solution to a Riemann-Hilbert problem G?₊=?₋ with ?₊ and ?₋ satisfying the corona conditions in C⁺ and C⁻, respectively. However, determining such a solution and verifying that the corona conditions are satisfied are in general difficult problems. In this paper, on one hand, we establish conditions on ?_± which are equivalent to the corona conditions but easier to verify, if G±1 are analytic and bounded in a strip. This happens in particular with almost-periodic symbols. On the other hand, we identify new classes of symbols G for which a non-trivial solution to G?₊=?₋ can be explicitly determined and the corona conditions can be verified by the above mentioned approach, thus obtaining invertibility criteria for the associated Toeplitz operators.     

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Let T∈Bn(H) be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H, and let AT⊂B(H) be the unital dual operator algebra generated by T. In this note we show that every operator S∈B(H) in the essential commutant of AT has the form S=X+K with a T-Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T=(Mz₁,…,Mzn)∈B(H²n(σ)) consisting of the multiplication operators with the coordinate functions on the Hardy space H²(σ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D⊂Cn. As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H²(σ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.