scholarly journals Block Toeplitz operators with frequency-modulated semi-almost periodic symbols

2003 ◽  
Vol 2003 (34) ◽  
pp. 2157-2176 ◽  
Author(s):  
A. Böttcher ◽  
S. Grudsky ◽  
I. Spitkovsky
Keyword(s):  
Toeplitz Operator ◽  
Frequency Modulation ◽  
Real Line ◽  
Matrix Function ◽  
Toeplitz Operators ◽  
Matrix Functions ◽  
Almost Periodic ◽  
The Matrix ◽  
Matrix Symbol ◽  
Block Toeplitz Operators

This paper is concerned with the influence of frequency modulation on the semi-Fredholm properties of Toeplitz operators with oscillating matrix symbols. The main results give conditions on an orientation-preserving homeomorphismαof the real line that ensure the following: ifbbelongs to a certain class of oscillating matrix functions (periodic, almost periodic, or semi-almost periodic matrix functions) and the Toeplitz operator generated by the matrix functionb(x)is semi-Fredholm, then the Toeplitz operator with the matrix symbolb(α(x))is also semi-Fredholm.

scholarly journals Vector Semi-Fredholm Toeplitz Operators and Mean Winding Numbers

2009 ◽  
Vol 195 ◽  
pp. 57-75
Author(s):  
Dmitry Yakubovich
Keyword(s):  
Toeplitz Operator ◽  
Real Line ◽  
Matrix Function ◽  
Toeplitz Operators ◽  
Necessary Conditions ◽  
Winding Number ◽  
The Real ◽  
The Matrix ◽  
Complex Valued

AbstractFor a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

Approximation of Irrational Matrix Functions and Its Application to Continuous-Discrete Model Conversion

1989 ◽  
Vol 111 (2) ◽  
pp. 142-145 ◽  
Author(s):  
Muh-Yang Chen ◽  
Chyi Hwang
Keyword(s):  
Matrix Function ◽  
Discrete Model ◽  
Matrix Functions ◽  
Rational Matrix ◽  
Improved Method ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Model Conversion ◽  
Irrational Function ◽  
Rational Matrix Function

In this paper, an improved method of rational approximation is presented for evaluating the irrational matrix function f(A), where A is a square matrix and f(s) is a scalar irrational function which is analytic on the spectrum of A. The improvement in the accuracy of the approximation off (A) by a rational matrix function is achieved by using the multipoint Pade approximants to f(s). An application example to model conversion involving the evaluations of the matrix exponential exp (AT) and the matrix logarithm ln(F) is provided to illustrate the superiority of the method.

scholarly journals Regular approximate factorization of a class of matrix-function with an unstable set of partial indices

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170279 ◽  
Author(s):  
G. Mishuris ◽  
S. Rogosin
Keyword(s):  
Matrix Function ◽  
Stable Set ◽  
Matrix Functions ◽  
Stable Sets ◽  
Approximate Factorization ◽  
Original Matrix ◽  
Partial Indices ◽  
Unstable Set ◽  
The Matrix ◽  
The Difference

From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk. XIII , 3–72. (Russian).), it is well known that the set of partial indices of a non-singular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions.

scholarly journals An effective recursive formula for the Frobenius covariants in matrix functions

2017 ◽  
Vol 5 (1) ◽  
pp. 113-122
Author(s):  
F. Schäfer
Keyword(s):  
Explicit Expression ◽  
Matrix Function ◽  
Recursive Formula ◽  
Matrix Exponential ◽  
Theoretical Studies ◽  
Matrix Functions ◽  
The Matrix ◽  
Frobenius Covariants

Abstract For theoretical studies, it is helpful to have an explicit expression for a matrix function. Several methods have been used to determine the required Frobenius covariants. This paper presents a recursive formula that calculates these covariants effectively. The new aspect of this method is the simple determination of the occurring coefficients in the covariants. The advantage is shown by several examples for the matrix exponential in comparision with Mathematica. The calculations are performed exactly.

scholarly journals The Nevanlinna parametrization for a matrix moment problem

2001 ◽  
Vol 89 (2) ◽  
pp. 245 ◽  
Author(s):  
Pedro Lopez-Rodriguez
Keyword(s):  
Half Plane ◽  
Moment Problem ◽  
Matrix Function ◽  
Unitary Matrix ◽  
Matrix Functions ◽  
Matrix Polynomials ◽  
Analytic Matrix ◽  
The Matrix ◽  
Upper Half Plane ◽  
Matrix Moment

We obtain the Nevanlinna parametrization for an indeterminate matrix moment problem, giving a homeomorphism between the set $V$ of solutions to the matrix moment problem and the set $\mathcal V$ of analytic matrix functions in the upper half plane such that $V(\lambda )^*V(\lambda )\le I$. We characterize the N-extremal matrices of measures (those for which the space of matrix polynomials is dense in their $L^2$-space) as those whose corresponding matrix function $V(\lambda )$ is a constant unitary matrix.

scholarly journals On the Matrix Versions of Incomplete Extended Gamma and Beta Functions and Their Applications for the Incomplete Bessel Matrix Functions

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Chaojun Zou ◽  
Mimi Yu ◽  
Ahmed Bakhet ◽  
Fuli He
Keyword(s):  
Matrix Function ◽  
Matrix Functions ◽  
The Matrix

In this paper, we first introduce the incomplete extended Gamma and Beta functions with matrix parameters; then, we establish some different properties for these new extensions. Furthermore, we give a specific application for the incomplete Bessel matrix function by using incomplete extended Gamma and Beta functions; at last, we construct the relation between the incomplete confluent hypergeometric matrix functions and incomplete Bessel matrix function.

Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions

1977 ◽  
Vol 29 (5) ◽  
pp. 937-946
Author(s):  
Hock Ong
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Transformation ◽  
Matrix Function ◽  
Multiplicative Group ◽  
Matrix Functions ◽  
Linear Transformations ◽  
General Matrix ◽  
The Matrix ◽  
Image Position

Let F be a field, F* be its multiplicative group and Mn(F) be the vector space of all n-square matrices over F. Let Sn be the symmetric group acting on the set {1, 2, … , n}. If G is a subgroup of Sn and λ is a function on G with values in F, then the matrix function associated with G and X, denoted by Gλ, is defined byand letℐ(G, λ) = { T : T is a linear transformation of Mn(F) to itself and Gλ(T(X)) = Gλ(X) for all X}.

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