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Chang Eon Shin

Publications and source records attributed to Chang Eon Shin.

6 recordsLinked to original sources
arXiv2025

On Sylvester equations in Banach subalgebras

Let ${\mathcal B}$ be a Banach algebra and ${\mathcal A}$ be a Banach subalgebra that admits norm-controlled inversion in ${\mathcal B}$. In this work, we take $A, B$ in the Banach subalgebra ${\mathcal A}$ with their spectra in the Banach algebra ${\mathcal B}$ being disjoint, and show that the operator Sylvester equation $ BX-XA=Q$ has a unique solution $X\in {\mathcal A}$ for every $Q\in {\mathcal A}$. Under the additional assumptions that ${\mathcal B}$ is the operator algebra ${\mathcal B}(H)$ on a Hilbert space $H$ and that $A$ and $B$ are normal in ${\mathcal B}(H)$, an explicit norm estimate for the solution $X$ of the above operator Sylvester equation is provided in this work. In addition, the above conclusion on norm control is applied to Banach subalgebras of localized infinite matrices and integral operators.

math.FA↗
arXiv2019

Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs

The (un)weighted stability for some matrices is one of essential hypotheses in time-frequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weights. Inverse-closed Banach subalgebras of matrices with certain off-diagonal decay can be informally interpreted as localization preservation under inversion, which is of great importance in many mathematical and engineering fields. Let ${\mathcal B}(\ell^p_w)$ be the Banach algebra of bounded operators on the weighted sequence space $\ell^p_w$ on a simple graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a simple graph are inverse-closed in ${\mathcal B}(\ell^p_w)$ for all $1\le p<\infty$ and Muckenhoupt $A_p$-weights $w$, and the Beurling norm of the inversion of a matrix $A$ is bounded by a bivariate polynomial of the Beurling norm of the matrix $A$ and the operator norm of its inverse $A^{-1}$ in ${\mathcal B}(\ell^p_w)$.

math.FA↗
arXiv2017

Polynomial control on stability, inversion and powers of matrices on simple graphs

Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel % Banach algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of ${\mathcal B}(\ell^p), 1\le p\le \infty$, the space of all bounded operators on the space $\ell^p$ of all $p$-summable sequences. The $\ell^p$-stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among $\ell^p$-stabilities of matrices in Beurling algebras for different exponents $1\le p\le \infty$, with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of ${\mathcal B}(\ell^2)$ have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network.

math.FA↗
arXiv2011

Stability of Localized Integral Operators on Weighted $L^p$ spaces

In this paper, we consider localized integral operators whose kernels have mild singularity near the diagonal and certain Holder regularity and decay off the diagonal. Our model example is the Bessel potential operator ${\mathcal J}_γ, γ>0$. We show that if such a localized integral operator has stability on a weighted function space $L^p_w$ for some $p\in [1, \infty)$ and Muckenhoupt $A_p$-weight $w$, then it has stability on weighted function spaces $L^{p'}_{w'}$ for all $1\le p'<\infty$ and Muckenhoupt $A_{p'}$-weights $w'$.

math.FA↗
arXiv2008

Stability of Localized Operators

Let $\ell^p, 1\le p\le \infty$, be the space of all $p$-summable sequences and $C_a$ be the convolution operator associated with a summable sequence $a$. It is known that the $\ell^p$- stability of the convolution operator $C_a$ for different $1\le p\le \infty$ are equivalent to each other, i.e., if $C_a$ has $\ell^p$-stability for some $1\le p\le \infty$ then $C_a$ has $\ell^q$-stability for all $1\le q\le \infty$. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the $\ell^p$- stability (or $L^p$-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.

math.FA↗