Invertibility of random matrices: norm of the inverse

On pseudo B-Weyl operators and generalized Drazin invertibility for operator matrices

We introduce a new class which generalizes the class of B-Weyl operators. We say that $T\in L(X)$ is pseudo B-Weyl if $T=T_1\oplus T_2$ where $T_1$ is a Weyl operator and $T_2$ is a quasi-nilpotent operator. We show that the corresponding pseudo B-Weyl spectrum $σ_{pBW}(T)$ satisfies the equality $σ_{pBW}(T)\cup[{\mathcal S}(T)\cap{\mathcal S}(T^*)]=σ_{gD}(T);$ where $σ_{gD}(T)$ is the generalized Drazin spectrum of $T\in L(X)$ and ${\mathcal S}(T)$ (resp., ${\mathcal S} (T^*)$) is the set where $T$ (resp., $T^*$) fails to have SVEP. We also investigate the generalized Drazin invertibility of …

Generalized Drazin-Riesz Invertibility for Operators Matrices

Let $A\in\mathcal{B}(X)$, $B\in\mathcal{B}(Y)$ and $C\in\mathcal{B}(Y,X)$ where $X$ and $Y$ are infinite Banach or Hilbert spaces. Let $M_{C}=\begin{pmatrix} A & C\cr 0 & B \end{pmatrix}$ be $2\times 2$ upper triangular operator matrix acting on $X\oplus Y$. In this paper, we consider some necessary and sufficient conditions for $M_{C}$ to be generalized Drazin-Riesz invertible. Furthermore, the set $\bigcap_{C\in \mathcal{B}(Y,X)}σ_{gDR}(M_{C})$ will be investigated and their relation between $\bigcap_{C\in \mathcal{B}(Y,X)}σ_{b}(M_{C})$ will be studied, where $σ_{gDR}(M_{C})$ and $σ_{b}(M_{C…

On the invertibility of "rectangular" bi-infinite matrices and applications in time--frequency analysis

Finite dimensional matrices having more columns than rows have no left inverses while those having more rows than columns have no right inverses. We give generalizations of these simple facts to bi--infinite matrices and use those to obtain density results for $p$--frames of time--frequency molecules in modulation spaces and identifiability results for operators with bandlimited Kohn--Nirenberg symbols.

希尔伯特空间上算子矩阵的可逆性性质

Invertibility properties of operator matrices on Hilbert spaces

本文针对希尔伯特空间上对角元已知的$n$维上三角算子矩阵$T_n^d(A)$，给出其各类Fredholm和Weyl补全的充要条件，修正前人结果并推广方法至任意$n\geq2$。

On the restricted invertibility problem with an additional orthogonality constraint for random matrices

The Restricted Invertibility problem is the problem of selecting the largest subset of columns of a given matrix $X$, while keeping the smallest singular value of the extracted submatrix above a certain threshold. In this paper, we address this problem in the simpler case where $X$ is a random matrix but with the additional constraint that the selected columns be almost orthogonal to a given vector $v$. Our main result is a lower bound on the number of columns we can extract from a normalized i.i.d. Gaussian matrix for the worst $v$.

Invertibility of symmetric random matrices

We study n by n symmetric random matrices H, possibly discrete, with iid above-diagonal entries. We show that H is singular with probability at most exp(-n^c), and the spectral norm of the inverse of H is O(sqrt{n}). Furthermore, the spectrum of H is delocalized on the optimal scale o(n^{-1/2}). These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of Tao and Vu, and Erdos, Schlein and Yau.

Invertibility of matrix type operators of infinite order with exponential off-diagonal decay

We analyse various exponential off-diagonal decay rates of the elements of infinite matrices and their inverses. It is known that such decay of the elements of an infinite matrix does not imply inverse--closeness, i.e. the inverse, if exists, does not have the same order of decay. We discuss some consequences and extensions of this result.

The Littlewood-Offord Problem and invertibility of random matrices

We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables X_k and real numbers a_k, determine the probability P that the sum of a_k X_k lies near some number v. For arbitrary coefficients a_k …

C*-algebra positive element invertibility criteria in terms of $L_1$-norms equivalence

We prove that the $L_1$-norms associated with a positive element $a$ of a unital C*-algebra are equivalent to the norm of C*-algebra if and only if $a$ is invertible.

上三角算子矩阵及其各种谱的稳定性

Upper triangular operator matrices and stability of their various spectra

本文研究了上三角算子矩阵的各类谱稳定性，提供了不同可逆性类型的特征化及稳定性条件，推广了前人结果并完善了相关理论体系。

Hirano inverse of anti-triangular matrix over Banach Algebras

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.

论Banach代数中的广义core-EP可逆性

On generalized core-EP invertibility in a Banach algebra

本文在Banach *-代数中研究广义core-EP逆，利用零化子刻画其性质，得到乘积的广义core-EP逆，并研究core-EP序及其在分块复矩阵中的应用。

Invertibility threshold for $H^\infty$ trace algebras, and effective matrix inversions

For a given $δ$, $0<δ<1$, a Blaschke sequence $σ=\{λ_j\}$ is constructed such that every function $f$, $f\in H^\infty$, having $δ<δ_f=\inf_{λ\inσ}|f(λ)|\le\|f\|_\infty\le1$ is invertible in the trace algebra $H^\infty|σ$ (with a norm estimate of the inverse depending on $δ_f$ only), but there exists $f$ with $δ=δ_f\le\|f\|_\infty\le1$, which does not. As an application, a counterexample to a stronger form of the Bourgain--Tzafriri restricted invertibility conjecture for bounded operators is exhibited, where an ``orthogonal (or unconditional) basis'' is replaced by a ``summation block orthogona…

A constructive inversion framework for twisted convolution

In this paper we develop constructive invertibility conditions for the twisted convolution. Our approach is based on splitting the twisted convolution with rational parameters into a finite number of weighted convolutions, which can be interpreted as another twisted convolution on a finite cyclic group. In analogy with the twisted convolution of finite discrete signals, we derive an anti-homomorphism between the sequence space and a suitable matrix algebra which preserves the algebraic structure. In this way, the problem reduces to the analysis of finite matrices whose entries are sequences su…

Left inverses of matrices with polynomial decay

The algebra of Schur operators on l^2 is known not to be inverse-closed. When l^2=l^2(X) where X is a metric space, we can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra for a strictly polynomial weight is inverse-closed. Here, we prove a result dealing with left-invertibility. Namely, if such an operator is bounded below in l^p for some p, then it is bounded below for all q, and it admits a left-inverse in the weighted Schur algebra.

Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices

In the first half of this text we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch & Silbermann and Lindner) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang. We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator $A$. In the second half of this text we study bounded linear operators on the generalised sequence space $\ell^p(\Z^N,U)$, where $p\in [1,\inf…

Restricted invertibility of continuous matrix functions

Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:\mathbb{R}\to M_{n\times n}$ and lower $\ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for any such $A$ with unit-length columns, there exists a continuous choice of subspaces $t\mapsto U(t)\subset \mathbb{R}^n$ such that for $v\in U(t)$, $\|A(t)v\|\geq c\|v\|$ where $c$ is some universal constant. Furthermore, the $U(t)$ are chosen so that their dimension satisfies a lower bound with optimal asymptotic dependence on $n$ and $\sup_{t\in \mathbb{R}}\…

Banach代数中群逆的加性结果

Additive results of group inverses in Banach algebras

本文研究Banach代数中两个群可逆元素之和的群逆表示，并将所得结果应用于分块复矩阵，从而得到某些分块复矩阵的群可逆性。

Restricted invertibility revisited

Suppose that $m,n\in \mathbb{N}$ and that $A:\mathbb{R}^m\to \mathbb{R}^n$ is a linear operator. It is shown here that if $k,r\in \mathbb{N}$ satisfy $k<r\le \mathrm{\bf rank(A)}$ then there exists a subset $σ\subseteq \{1,\ldots,m\}$ with $|σ|=k$ such that the restriction of $A$ to $\mathbb{R}^σ\subseteq \mathbb{R}^m$ is invertible, and moreover the operator norm of the inverse $A^{-1}:A(\mathbb{R}^σ)\to \mathbb{R}^m$ is at most a constant multiple of the quantity $\sqrt{mr/((r-k)\sum_{i=r}^m \mathsf{s}_i(A)^2)}$, where $\mathsf{s}_1(A)\geqslant\ldots\geqslant \mathsf{s}_m(A)$ are the singula…

任意Banach空间上3×3上三角算子矩阵的补全问题

Completion problem of upper triangular $3\times3$ operator matrices on arbitrary Banach spaces

本文研究了任意Banach空间上3×3上三角算子矩阵的补全问题，利用Harte的指数定理和Banach空间嵌入技术，推广了Han等人结论，并探讨了高维上三角算子的可逆性条件。

The inverse function theorem and the resolution of the Jacobian conjecture in free analysis

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the function must be invertible. Thus, as a corollary, we establish the Jacobian conjecture in this context. Furthermore, our result holds for commutative polynomials evaluated on tuples of commuting matrices.

A note on $l^2$ norms of weighted mean matrices

We give a proof of Cartlidge's result on the $l^{p}$ operator norms of weighted mean matrices for $p=2$ on interpreting the norms as eigenvalues of certain matrices.

On the Expectation of the Norm of Random Matrices with Non-Identically Distributed

We give estimates for the expectation of the norm of random matrices with independent but not necessarily identically distributed entries.

On $l^p$ norms of factorable matrices

We study $l^p$ operator norms of factorable matrices and related results. We give applications to $l^p$ operator norms of weighted mean matrices and Copson's inequalities. We also apply the method in this paper to study the best constant in an inequality of Hardy, Littlewood and Pólya.

On l^p norms of weighted mean matrices

We study $l^{p}$ operator norms of weighted mean matrices using the approaches of Kaluza-Szegö and Redheffer. As an application, we prove a conjecture of Bennett.

A note on $l^p$ norms of weighted mean matrices

We present some results concerning the $l^p$ norms of weighted mean matrices. These results can be regarded as analogues to a result of Bennett concerning weighted Carleman's inequalities.

On $l^2$ norms of some weighted mean matrices

We give another proof of a result of Bennett on the $l^{p}$ operator norms of some weighted mean matrices for the case $p=2$ and we also present some related results.

Norm and numerical radius inequalities for operator matrices

Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices whose real and imaginary parts are positive will be discussed, and sharper bounds will be shown for such class.

The $l^p$ norms of a class of weighted mean matrices

We study the $l^p$ norms of a class of weighted mean matrices whose diagonal terms are given by $n^α/\sum^{n}_{i=1}i^α$ with $α> -1$. The $l^p$ norms of such matrices are known for $p \geq 2, (α+1)p >1$ and $1<p \leq 4/3, 1/p \leq α\leq 1$. In this paper, we determine the $l^p$ norms of such matrices for $p \geq 1.35, 0\leq α\leq 1$.

The p-norm of some matrices

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 $p\in [1,\infty]$.

On weighted mean matrices whose $l^p$ norms are determined on decreasing sequences

We give a condition on weighted mean matrices so that their $l^p$ norms are determined on decreasing sequences when the condition is satisfied. We apply our result to give a proof of a conjecture of Bennett and discuss some related results.

The Normal Defect of Some Classes of Matrices

An n \times n matrix A has a normal defect of k if there exists an (n+k) \times (n+k) normal matrix A_{ext} with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4 \times 4 matrices, namely matrices whose only nonzero entries lie on the superdiagonal, and we provide details for constructing minimal normal completion matrices A_{ext}. We also prove a result for a related class of n \times n matrices. Finally, we present an example of a 6 \times 6 block diagonal matrix having the property that its normal defect is strictly less th…

Schur multiplier norms for Loewner matrices

We study upper bounds on the Schur multiplier norm of Loewner matrices for concave and convex functions. These bounds then immediately lead to upper bounds on the ratio of Schatten $q$-norms of commutators $||\;[A,f(B)]\;||_q/||\;[A,B]\;||_q$. We also consider operator monotone functions, for which sharper bounds are obtained.

Coburn's Lemma 和随机Jacobi算子的有限截断方法

Coburn's Lemma and the Finite Section Method for Random Jacobi Operators

本文研究了有限和无限三对角随机矩阵的谱和伪谱，证明了在伪ergodic条件下，有限和无限三对角随机矩阵的谱具有特定性质，并展示了有限截断方法在求解方程和估计谱时的有效性。

Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehman-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.

On Symmetric Norm Inequalities And Hermitian Block-Matrices

The main purpose of this paper is to englobe some new and known types of Hermitian block-matrices $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix}$ satisfying or not the inequality $\|M\|\le \|A+B\|$ for all symmetric norms

Concentration of norms and eigenvalues of random matrices

We prove concentration results for $\ell_p^n$ operator norms of rectangular random matrices and eigenvalues of self-adjoint random matrices. The random matrices we consider have bounded entries which are independent, up to a possible self-adjointness constraint. Our results are based on an isoperimetric inequality for product spaces due to Talagrand.

Norms on Complex Matrices Induced by Random Vectors

We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter's positivity theorem for the complete homogeneous symmetric polynomials.