1,763 research outputs found
Cowen's class and Thomson's class
In studying commutants of analytic Toeplitz operators, Thomson proved a
remarkable theorem which states that under a mild condition, the commutant of
an analytic Toeplitz operator is equal to that of Toeplitz operator defined by
a finite Blaschke product. Cowen gave an significant improvement of Thosom's
result. In this paper, we will present examples in Cowen's class which does not
lie in Thomson's class.Comment: 14 pages, 2 figure
Product of truncated Hankel and truncated Toeplitz operators
A truncated Toeplitz operator is the compression of a classical Toeplitz
operator on the Hardy space to a model space. A truncated Hankel operator is
the compression of a Hankel operator on the Hardy space to the orthogonal
complement of a model space. We study the product of a truncated Hankel
operator and a truncated Toeplitz operator, and characterize when such a
product is zero or compact
Asymptotic equivariant index of Toeplitz operators and relative index of CR structures
Using equivariant Toeplitz operator calculus, we give a new proof of the
Atiyah-Weinstein conjecture on the index of Fourier integral operators and the
relative index of CR structures.Comment: 23 page
Compact Product of Hankel and Toeplitz Operators
In this paper, we study the product of a Hankel operator and a Toeplitz
operator on the Hardy space. We give necessary and sufficient conditions of
when such a product is compact
The restriction operator on Bergman spaces
We study the restriction operator from the Bergman space of a domain in
to the Bergman space of a non-empty open subset of the domain.
We relate the restriction operator to the Toeplitz operator on the Bergman
space of the domain whose symbol is the characteristic function of the subset.
Using the biholomorphic invariance of the spectrum of the associated Toeplitz
operator, we study the restriction operator from the Bergman space of the unit
disc to the Bergman space of subdomains with large symmetry groups, such as
horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp
estimate of the norm of the restriction operator in case the domain and the
subdomain are balls. We also study various operator theoretic properties of the
restriction operator such as compactness and essential norm estimates.Comment: Reference to previous work on restriction operators is adde
Matrix-valued truncated Toeplitz operators: unbounded symbols, kernels and equivalence after extension
This paper studies matrix-valued truncated Toeplitz operators, which are a
vectorial generalisation of truncated Toeplitz operators. It is demonstrated
that, although there exist matrix-valued truncated Toeplitz operators without a
matrix symbol in for any , there is a wide class of
matrix-valued truncated Toeplitz operators which possess a matrix symbol in
for some . In the case when the matrix-valued
truncated Toeplitz operator has a symbol in for some , an approach is developed which bypasses some of the technical difficulties
which arise when dealing with problems concerning matrix-valued truncated
Toeplitz operators with unbounded symbols. Using this new approach, two new
notable results are obtained. The kernel of the matrix-valued truncated
Toeplitz operator is expressed as an isometric image of an -invariant
subspace. Also, a Toeplitz operator is constructed which is equivalent after
extension to the matrix-valued truncated Toeplitz operator
Time-Limited Toeplitz Operators on Abelian Groups: Applications in Information Theory and Subspace Approximation
Toeplitz operators are fundamental and ubiquitous in signal processing and
information theory as models for linear, time-invariant (LTI) systems. Due to
the fact that any practical system can access only signals of finite duration,
time-limited restrictions of Toeplitz operators are naturally of interest. To
provide a unifying treatment of such systems working on different signal
domains, we consider time-limited Toeplitz operators on locally compact abelian
groups with the aid of the Fourier transform on these groups. In particular, we
survey existing results concerning the relationship between the spectrum of a
time-limited Toeplitz operator and the spectrum of the corresponding
non-time-limited Toeplitz operator. We also develop new results specifically
concerning the eigenvalues of time-frequency limiting operators on locally
compact abelian groups. Applications of our unifying treatment are discussed in
relation to channel capacity and in relation to representation and
approximation of signals
Toeplitz operators on doubling Fock spaces
We study Toeplitz operator theory on the doubling Fock spaces, which are Fock
spaces whose exponential weight is associated to a subharmonic function with
doubling Riesz measure. Namely, we characterize the boundedness, compactness
and membership in the Schatten class of Toeplitz operators on doubling Fock
spaces whose symbol is a positive Radon measure
Slant H-Toeplitz Operators on the Hardy space
The notion of slant H-Toeplitz operator on the Hardy space is
introduced and its characterizations are obtained. We have shown that an
operator on the space is slant H-Toeplitz if and only if its matrix is a
slant H-Toeplitz matrix. In addition the conditions under which slant Toeplitz
and slant Hankel operators become slant H-Toeplitz operators are also obtained
The Sarason Sub-Symbol and the Recovery of the Symbol of Densely Defined Toeplitz Operators over the Hardy Space
While the symbol map for the collection of bounded Toeplitz operators is well
studied, there has been little work on a symbol map for densely defined
Toeplitz operators. In this work a family of candidate symbols, the Sarason
Sub-Symbols, is introduced as a means of reproducing the symbol of a densely
defined Toeplitz operator. This leads to a partial answer to a question posed
by Donald Sarason in 2008. In the bounded case the Toeplitzness of an operator
can be classified in terms of its Sarason Sub-Symbols. This justifies the
investigation into the application of the Sarason Sub-Symbols on densely
defined operators. It is shown that analytic closed densely defined Toeplitz
operators are completely determined by their Sarason Sub-Symbols, and it is
shown for a broader class of operators that they extend closed densely defined
Toeplitz operators (of multiplication type).Comment: 14 page
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