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Review: On Complex Symmetric Toeplitz Operators, Stephan Ramon Garcia
Review: On Complex Symmetric Toeplitz Operators, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin
An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin
Pomona Faculty Publications and Research
Suppose E is a subset of the unit circle T and Hinfinity C Linfinity is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to znHinfinity. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.
On The Norm Closure Problem For Complex Symmetric Operators, Stephan Ramon Garcia, Daniel E. Poore '11
On The Norm Closure Problem For Complex Symmetric Operators, Stephan Ramon Garcia, Daniel E. Poore '11
Pomona Faculty Publications and Research
We prove that the set of all complex symmetric operators on a separable, infinite-dimensional Hilbert space is not norm closed.
Unitary Equivalence To A Truncated Toeplitz Operator: Analytic Symbols, Stephan Ramon Garcia, Daniel E. Poore '11, William T. Ross
Unitary Equivalence To A Truncated Toeplitz Operator: Analytic Symbols, Stephan Ramon Garcia, Daniel E. Poore '11, William T. Ross
Pomona Faculty Publications and Research
Unlike Toeplitz operators on H², truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this paper we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive, and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension ≤ 3 is unitarily equivalent to a direct sum of truncated Toeplitz operators.
Unitary Equivalence To A Complex Symmetric Matrix: Low Dimensions, Stephan Ramon Garcia, Daniel E. Poore '11, James E. Tener '08
Unitary Equivalence To A Complex Symmetric Matrix: Low Dimensions, Stephan Ramon Garcia, Daniel E. Poore '11, James E. Tener '08
Pomona Faculty Publications and Research
A matrix T∈Mn(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above.
On The Closure Of The Complex Symmetric Operators: Compact Operators And Weighted Shifts, Stephan Ramon Garcia, Daniel E. Poore '11
On The Closure Of The Complex Symmetric Operators: Compact Operators And Weighted Shifts, Stephan Ramon Garcia, Daniel E. Poore '11
Pomona Faculty Publications and Research
We study the closure $\bar{CSO}$ of the set $CSO$ of all complex symmetric operators on a separable, infinite-dimensional, complex Hilbert space. Among other things, we prove that every compact operator in $\bar{CSO}$ is complex symmetric. Using a construction of Kakutani as motivation, we also describe many properties of weighted shifts in $\bar{CSO} \backslash CSO$. In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to $\bar{CSO}\backslash CSO$. As a byproduct of our treatment of weighted shifts, we explain several ways in which our result on compact operators is optimal.
Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen
Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen
Pomona Faculty Publications and Research
We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ so that $T = CT^*C$. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data $(\dim \ker T, \dim \ker T^*)$.
Unitary Equivalence To A Complex Symmetric Matrix: Geometric Criteria, Levon Balayan '09, Stephan Ramon Garcia
Unitary Equivalence To A Complex Symmetric Matrix: Geometric Criteria, Levon Balayan '09, Stephan Ramon Garcia
Pomona Faculty Publications and Research
We develop several methods, based on the geometric relationship between the eigenspaces of a matrix and its adjoint, for determining whether a square matrix having distinct eigenvalues is unitarily equivalent to a complex symmetric matrix. Equivalently, we characterize those matrices having distinct eigenvalues which lie in the unitary orbit of the complex symmetric matrices.
The Norm Of A Truncated Toeplitz Operator, Stephan Ramon Garcia, William T. Ross
The Norm Of A Truncated Toeplitz Operator, Stephan Ramon Garcia, William T. Ross
Pomona Faculty Publications and Research
We prove several lower bounds for the norm of a truncated Toeplitz operator and obtain a curious relationship between the H2 and H∞ norms of functions in model spaces.
Complex Symmetric Partial Isometries, Stephan Ramon Garcia, Warren R. Wogen
Complex Symmetric Partial Isometries, Stephan Ramon Garcia, Warren R. Wogen
Pomona Faculty Publications and Research
An operator $T \in B(\h)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\h\to\h$ so that $T = CT^*C$. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension $\leq 4$ is complex symmetric.
Interpolation And Complex Symmetry, Stephan Ramon Garcia, Mihai Putinar
Interpolation And Complex Symmetry, Stephan Ramon Garcia, Mihai Putinar
Pomona Faculty Publications and Research
In a separable complex Hilbert space endowed with an isometric conjugate-linear involution, we study sequences orthonormal with respect to an associated bilinear form. Properties of such sequences are measured by a positive, possibly unbounded angle operator which is formally orthogonal as a matrix. Although developed in an abstract setting, this framework is relevant to a variety of eigenvector interpolation problems arising in function theory and in the study of differential operators.
Complex Symmetric Operators And Applications Ii, Stephan Ramon Garcia, Mihai Putinar
Complex Symmetric Operators And Applications Ii, Stephan Ramon Garcia, Mihai Putinar
Pomona Faculty Publications and Research
A bounded linear operator T on a complex Hilbert space H is called complex symmetric if T = CT*C, where C is a conjugation (an isometric, antilinear involution of H). We prove that T = CJ|T|, where J is an auxiliary conjugation commuting with |T| = √{T*T). We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition T = CJ|T| also extends to the class of unbounded C-self adjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms …