Subnormal Toeplitz Operators Research Articles

Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols

In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmos's Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic? We first prove: Let Φ∈LMn∞ be a matrix-valued rational function having a “matrix pole”, i.e., there exists α∈D for which kerHΦ⊆(z−α)HCn2, where HΦ denotes the Hankel operator with symbol Φ. If(i)TΦ is hyponormal;(ii)ker[TΦ⁎,TΦ] is invariant for TΦ, then TΦ is normal. Hence in particular, if TΦ is subnormal then TΦ is normal.Next, we show that every pure quasinormal Toeplitz operator with a matrix-valued rational symbol is unitarily equivalent to an analytic Toeplitz operator.

Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmosʼs Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamseʼs theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization”, is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well-known conjecture, of whether every subnormal Toeplitz operator with finite rank self-commutator is normal or analytic.

Hyponormality and subnormality of block Toeplitz operators

In this paper, we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space HCn2 of the unit circle.First, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator.Second, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos’s Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if Φ is a matrix-valued rational function whose co-analytic part has a coprime factorization then every hyponormal Toeplitz operator TΦ whose square is also hyponormal must be either normal or analytic.Third, using the subnormal theory of block Toeplitz operators, we give an answer to the following “Toeplitz completion” problem: find the unspecified Toeplitz entries of the partial block Toeplitz matrix A≔[U∗??U∗] so that A becomes subnormal, where U is the unilateral shift on H2.

Subnormal Toeplitz operators and the kernels of their self-commutators

In this note we give a connection between subnormal Toeplitz operators and the kernels of their self-commutators. This is closely related to P.R. Halmos's Problem 5: Is every subnormal Toeplitz operator either normal or analytic? Our main theorem is as follows: If φ ∈ L ∞ is such that φ and φ ¯ are of bounded type (that is, they are quotients of two analytic functions on the open unit disk) and if the kernel of the self-commutator of T φ is invariant for T φ then T φ is either normal or analytic.

Approximate Factorization in Generalized Hardy Spaces

In this paper we establish a connection between the approximate factorization property appearing in the theory of dual algebras and the spectral inclusion property for a class of Toeplitz operators on Hilbert spaces of vector valued square integrable functions. As an application, it follows that a wide range of dual algebras of subnormal Toeplitz operators on various Hardy spaces associated to function algebras have property (A1(1)). It is also proved that the dual algebra generated by a spherical isometry (with a possibly infinite number of components) has the same property. One particular application is given to the existence of unimodular functions sitting in cyclic invariant subspaces of weak* Dirichlet algebras. Moreover, by this method we provide a unified approach to several Toeplitz spectral inclusion theorems.

Hyponormal Toeplitz Operators and Extremal Problems of Hardy Spaces

The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos's question about a subnormal Toeplitz operator when the self-commutator is finite rank

Hyponormal Toeplitz operators and extremal problems of Hardy spaces

The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos’s question about a subnormal Toeplitz operator when the self-commutator is finite rank.

More subnormal Toeplitz operators.

More subnormal Toeplitz operators.

Note on ?the analytic model of a hyponormal operator with rank one self commutator?

The analytic model of a general hyponormal operator with rank one self-commutator is obtained. A Cauchy integral representation for the adjoint is given, certain kernels are discussed, and conditions are given under which the hyponormal operator is similar to a subnormal Toeplitz operator.

Some subnormal Toeplitz operators.

Some subnormal Toeplitz operators.

Subnormal Toeplitz operators and functions of bounded type

Subnormal Toeplitz operators and functions of bounded type

Subnormality and quasinormality of Toeplitz operators

Halmos asks whether every subnormal Toeplitz operator on H2 is either analytic or normal. It is shown that for a certain class of Toeplitz operators, the subnormality implies either analyticity or normality.