arXiv subjects
Publications and source records attributed to Muhammad Ahsan Khan.
The classification of maximal algebras of square block Toeplitz matrices is a considerably more difficult problem and has received relatively little attention in the existing literature. In this work, we approach the problem under the assumption that the entries belong to the Schur algebra. Within these settings, we obtain a complete classification of all maximal algebras of such block Toeplitz matrices.
Toeplitz matrices are characterized by their constant diagonals, have been extensively studied in various settings, including over real and complex numbers. However, their study over quaternions is quite sparse. In this paper, we investigate the structure and the algebraic properties of quaternion Toeplitz matrices. Most importantly, we established a complete characterization of all normal Toeplitz matrices having entries commutative quaternions.
Given $A,B,C$ and $D$ block Toeplitz matrices, we will prove some of the basic results concerning the product $AB-CD$. In addition, with respect to change of basis, the characterization of normal block Toeplitz matrices with entries in the algebra of diagonal matrices is also obtained.
In this paper, some features of countably $α$-compact topological spaces are presented and proven. The connection between countably $α$% -compact, Tychonoff, and $α$-Hausdorff spaces is explained. The space is countably $α$-compact space iff every locally finite family of non-empty subsets of such space is finite is demonstrated. The countably $% α$-compact space with weight greater than or equal to $\aleph_0$ is the $α$-continuous image of a closed subspace of the cube $D^{\aleph_0}$ is discussed. The boundedness of $α$-continuous functions mapping $α$% -compact spaces to other spaces is cleared. Moreover, the $α$% -continuous function mapping the space $X$ to the countably $α$-compact space $Y$ is an $α$-closed subset of $X\times Y$ is argued and proved. We explained that the $α$-continuous functions mapping any topological space to a countably $α$-compact space can be extended over its domain under some constraints. We claimed that the property of being $α$% -compact is countably $α$-compact but the converse is not and the countable union of countably $α$-compact subspaces of $X$ is also countably $α$-compact.
Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra $Å$. The characterization of normal block Toeplitz matrices with entries from $Å$ is also obtained.
In the last decade, a large amount of research has been concentrated on the operators living on the model space. Asymmetric truncated Toeplitz operators and asymmetric truncated Hankel operators are the natural generalization of truncated Toeplitz operators and truncated Hankel operators respectively. In this paper, we obtained the basic results concerning the product of these operators and in terms of product their connection with each other. In addition, when the inner function has a certain symmetric property, some algebraic properties of truncated Hankel operators are also discussed.
Let $A_Φ$ be a matrix valued truncated Toeplitz operator-the compression of multiplication operator to vector-valued model space $H^2(E)\ominus ΘH^2(E)$, where $Θ$ is a matrix valued non constant inner function. Under supplementary assumptions, we find necessary and sufficient condition that the product $A_ΦA_Ψ$ is itself a matrix valued truncated Toeplitz operator.
The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. The identification of algebras consisting of block Toeplitz matrices is a harder problem, that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra $ \mathcal{A} $. After obtaining some general results, we classify all the maximal algebras for certain particular cases of $ \mathcal{A}$.
The maximal commutative subalgebras containing only Toeplitz matrices have been identified as generalized circulants. A similar simple description cannot be obtained for block Toeplitz matrices. We introduce and investigate certain families of maximal commutative algebras of block Toeplitz matrices.