In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in $\ell^p$ for some $p$ implies boundedness below in $\ell^p$ for all $p$. We use the established resultto enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matrices.
We study exponential factorization of invertible matrices over unital complex Banach algebras. In particular, we prove that every invertible matrix with entries in the algebra of holomorphic functions on a closed bordered Riemann surface can be written as a product of two exponents of matrices over this algebra. Our result extends similar results proved earlier in [KS] and [L] for $2\times 2$ matrices.
In this paper, we present new necessary and sufficient conditions under which the sum of two group invertible elements in a Banach algebra has group inverse. We then apply these results to block operator matrices over Banach spaces. The group inverses of certain operator block matrices are thereby obtained. Additionally, this paper extends the results obtained in \cite{L}.
In this paper we investigate Hirano invertibility of anti-triangular matrix over a Banach algebra. Let $a\in {\mathcal A}^H, b\in {\mathcal A}^{sD}.$ If $b^Da=0, bab^π=0,$ we prove that $\begin{pmatrix} a&1\\ b&0 \end{pmatrix}\in M_2(\mathcal A)^H.$ Moreover, we considered Hirano invertibility of anti-triangular matrices under commutative-like conditions. These provide new kind of operator matrices with tripotent and nilpotent decompositions.
We investigate the Hirano invertibility of block-operator matrices in Banach algebras, and obtain the Hirano inverse of matrix $\begin{bmatrix} A&B\\ C&D \end{bmatrix}$ under two types of new perturbation conditions. Furthermore, we provide a new operator matrix decomposed into the sum of tripotent and nilpotent elements on Banach spaces.
For a commutative unital ring $R$, and $n\in \mathbb{N}$, let $\textrm{SL}_n(R)$ denote the special linear group over $R$, and $\textrm{E}_n(R)$ the subgroup of elementary matrices. Let ${\mathcal{M}}^+$ be the Banach algebra of all complex Borel measures on $[0,+\infty)$ with the norm given by the total variation, the usual operations of addition and scalar multiplication, and with convolution. It is shown that $\textrm{SL}_n(A)=\textrm{E}_n(A)$ for Banach subalgebras $A$ of ${\mathcal{M}}^+$ that are closed under the operation ${\mathcal{M}}^+\owns μ\mapsto μ_t$, $t\in [0,1]$, where $μ_t(E):…
In this paper, we present a new characterization of g-Drazin inverse in a Banach algebra. We prove that an element a is a Banach algebra has g-Drazin inverse if and only if there exists $x\in A$ such that $ax=xa, a-a^2x\in A^{qnil}$. we obtain the sufficient and necessary conditions for the existence of the g-Drain inverse for certain $2 \times 2$ anti-triangular matrices over a Banach algebra. These extend the results of Koliha (Glasgow Math. J., 38(1996), 367-381), Nicholson (Comm. Algebra,27(1999), 3583-3592 and Zou et al. (Studia Scient. Math. Hungar., 54(2017,489-508).
We compute the operator $p$-norm of some $n\times n$ complex matrices, which can be seen as bounded linear operators on the $n$ dimensional Banach space $\ell^p(n)$. The notion of logarithmic affine matrices is defined, and for such a matrix its $p$-norm is computed exactly. In particular, a matrix $A=\begin{pmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{pmatrix}$ which corresponds to a magic square belongs to the class of logarithmic affine matrices, and its $p$-norm is equal to $15$ for any $p\in [1,\infty]$.
The Generalized Locally Toeplitz (GLT) sequences of matrices have been originated from the study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate some partial differential equations by discretization. The study of the asymptotic spectral behaviour of GLT sequence is very important in analysing the solution of corresponding partial differential equations. The approximating classes of sequences (a.c.s) and the spectral symbols are important notions in this connection. Recently, G. Barbarino obtained some additional results…
We compute the operator $(p,q)$-norm of some $n\times n$ complex matrices, which can be seen as bounded linear operators from the $n$ dimensional Banach space $\ell^p(n)$ to $\ell^q(n)$. We have shown that a special matrix $A=\begin{pmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{pmatrix}$ which corresponds to a magic square has $\|A\|_{p,p} = \max \{\|Aξ\|_p : ξ\in\ell^p(n), \|ξ\|_p=1\} =15$ for any $p\in [1,\infty]$. In this paper, we extend this result and we compute $\|A\|_{p,q}$ for $1\le q \le p \le \infty$.
As a follow-up to work done in [7], some new insights to the structure of the socle of a semisimple Banach algebra is obtained. In particular, it is shown that the socle is isomorphic as an algebra to the direct sum of tensor products of corresponding left and right minimal ideals. Remarkably, the finite-dimensional case here reduces to the classical Wedderburn-Artin Theorem, and this approach does not use any continuous irreducible representations of the algebra in question. Furthermore, the structure of the socles for which the classical Shoda's Theorem for matrices can be extended, is chara…
In solving scientific, engineering or pure mathematical problems one is often faced with a need to approximate the function of a given class by the linear combination of a preferably small number of functions that are localised one way or another both in the time and frequency domain. Over the last seventy years or so a range of systems of thus localised functions have been developed to allow the decomposition and synthesis of functions of various classes. The most prominent examples of such systems are Gabor functions, wavelets, ridgelets, curvelets, shearlets and wave atoms. We recently intr…
Let $A\in \mathcal{B}(X)$ and $B\in \mathcal{B}(Y)$, where $X$ and $Y$ are Banach spaces, and let $M_{C}$ be an operator acting on $X\oplus Y$ given by $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}$. We investigate the limit point set of the Browder spectrum of $M_{C}$. It is shown that $$ acc σ_{b}(M_C)\cup W_{acc σ_{b}}= acc σ_{b}(A)\cup acc σ_{b}(B)$$ where $W_{acc σ_{b}}$ is a subsets of $ accσ_{*}(B)\cap accσ_{*}(A)$ and a union of certain holes in $ acc σ_{b}(M_C)$. Furthermore, several sufficient conditions for $accσ_{b}(M_C)=accσ_{b}(A)\cup accσ_{b}(B)$ holds for every $C\in \ma…
Let $(G,d)$ be a metric space with the counting measure $μ$ satisfying some growth conditions. Let $ω(x,y)=(1+d(x,y))^δ$ for some $0<δ\leq1$. Let $0<p\leq1$. Let $\mathcal A_{pω}$ be the collection of kernels $K$ on $G\times G$ satisfying $\max\{\sup_x\sum_y |K(x,y)|^pω(x,y)^p, \sup_y\sum_x |K(x,y)|^pω(x,y)^p\}<\infty$. Each $K \in \mathcal A_{pω}$ defines a bounded linear operator on $\ell^2(G)$. If in addition, $ω$ satisfies the weak growth condition, then we show that $\mathcal A_{pω}$ is inverse closed in $B(\ell^2(G))$. We shall also discuss inverse-closedness of $p$-Banach algebra of inf…
Rudelson's theorem states that if for a set of unit vectors $u_i$ and positive weights $c_i$, we have that $\sum c_i u_i\otimes u_i$ is the identity operator $I$ on ${\mathbb R}^d$, then the sum of a random sample of $Cd\ln d$ of these diadic products is close to $I$. The $\ln d$ term cannot be removed. On the other hand, the recent fundamental result of Batson, Spielman and Srivastava and its improvement by Marcus, Spielman and Srivastava show that the $\ln d$ term can be removed, if one wants to show the existence of a good approximation of $I$ as the average of a few diadic products. It is …
