Approximating real linear operators

Marko Huhtanen; Olavi Nevanlinna

Displaying similar documents to “Approximating real linear operators”

The Re-nonnegative definite solutions to the matrix equation $A X B = C$

Qing Wen Wang, Chang Lan Yang (1998)

Commentationes Mathematicae Universitatis Carolinae

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An $n \times n$ complex matrix $A$ is called Re-nonnegative definite (Re-nnd) if the real part of $x^{*} A x$ is nonnegative for every complex $n$ -vector $x$ . In this paper criteria for a partitioned matrix to be Re-nnd are given. A necessary and sufficient condition for the existence of and an expression for the Re-nnd solutions of the matrix equation $A X B = C$ are presented.

Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding

Jiří Rohn (2007)

Applications of Mathematics

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For a real square matrix $A$ and an integer $d \geq 0$ , let $A_{(d)}$ denote the matrix formed from $A$ by rounding off all its coefficients to $d$ decimal places. The main problem handled in this paper is the following: assuming that $A_{(d)}$ has some property, under what additional condition(s) can we be sure that the original matrix $A$ possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists...

On an extension of Fekete’s lemma

Inheung Chon (1999)

Czechoslovak Mathematical Journal

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We show that if a real $n \times n$ non-singular matrix ( $n \geq m$ ) has all its minors of order $m - 1$ non-negative and has all its minors of order $m$ which come from consecutive rows non-negative, then all $m$ th order minors are non-negative, which may be considered an extension of Fekete’s lemma.

Young's (in)equality for compact operators

Gabriel Larotonda (2016)

Studia Mathematica

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If a,b are n × n matrices, T. Ando proved that Young’s inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then $λ_{k} (| a b * |) \leq λ_{k} {(1 / p | a |}^{p} + 1 / {q | b |}^{q})$ for all k. Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young’s inequality if and only if ${| a |}^{p} = {| b |}^{q}$ .

On the matrix negative Pell equation

Aleksander Grytczuk, Izabela Kurzydło (2009)

Discussiones Mathematicae - General Algebra and Applications

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Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) $\sum_{i = 1}^{n} X ₂_{i} - d \sum_{i = 1}^{n} Y ²_{i} = - I$ with d ∈ N for nonsingular $X_{i}, Y_{i} \in M ₂ (Z)$ , i=1,...,n.

Disjoint strict singularity of inclusions between rearrangement invariant spaces

Francisco L. Hernández, Víctor M. Sánchez, Evgueni M. Semenov (2001)

Studia Mathematica

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It is studied when inclusions between rearrangement invariant function spaces on the interval [0,∞) are disjointly strictly singular operators. In particular suitable criteria, in terms of the fundamental function, for the inclusions $L ¹ \cap L^{\infty} ↪ E$ and $E ↪ L ¹ + L^{\infty}$ to be disjointly strictly singular are shown. Applications to the classes of Lorentz and Marcinkiewicz spaces are given.

Finite-rank perturbations of positive operators and isometries

Man-Duen Choi, Pei Yuan Wu (2006)

Studia Mathematica

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We completely characterize the ranks of A - B and $A^{1 / 2} - B^{1 / 2}$ for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m = r a n k (A^{1 / 2} - B^{1 / 2})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that...

On block triangular matrices with signed Drazin inverse

Changjiang Bu, Wenzhe Wang, Jiang Zhou, Lizhu Sun (2014)

Czechoslovak Mathematical Journal

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The sign pattern of a real matrix $A$ , denoted by $sgn A$ , is the $(+, -, 0)$ -matrix obtained from $A$ by replacing each entry by its sign. Let $𝒬 (A)$ denote the set of all real matrices $B$ such that $sgn B = sgn A$ . For a square real matrix $A$ , the Drazin inverse of $A$ is the unique real matrix $X$ such that $A^{k + 1} X = A^{k}$ , $X A X = X$ and $A X = X A$ , where $k$ is the Drazin index of $A$ . We say that $A$ has signed Drazin inverse if $sgn {\tilde{A}}^{d} = sgn A^{d}$ for any $\tilde{A} \in 𝒬 (A)$ , where $A^{d}$ denotes the Drazin inverse of $A$ . In this paper, we give necessary conditions for some block triangular matrices...

Rational realization of the minimum ranks of nonnegative sign pattern matrices

Wei Fang, Wei Gao, Yubin Gao, Fei Gong, Guangming Jing, Zhongshan Li, Yan Ling Shao, Lihua Zhang (2016)

Czechoslovak Mathematical Journal

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A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set ${+, -, 0}$ ( ${+, 0}$ , respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix $𝒜$ is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of $𝒜$ . Using a correspondence between sign patterns with minimum rank $r \geq 2$ and point-hyperplane configurations in $ℝ^{r - 1}$ and Steinitz’s theorem on the rational realizability...

Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces

Sergei V. Astashkin, Francisco L. Hernández, Evgeni M. Semenov (2009)

Studia Mathematica

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If G is the closure of $L_{\infty}$ in exp L₂, it is proved that the inclusion between rearrangement invariant spaces E ⊂ F is strictly singular if and only if it is disjointly strictly singular and E ⊊ G. For any Marcinkiewicz space M(φ) ⊂ G such that M(φ) is not an interpolation space between $L_{\infty}$ and G it is proved that there exists another Marcinkiewicz space M(ψ) ⊊ M(φ) with the property that the M(ψ) and M(φ) norms are equivalent on the Rademacher subspace. Applications are given and a question...

On the differentiability of certain saltus functions

Gerald Kuba (2011)

Colloquium Mathematicae

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We investigate several natural questions on the differentiability of certain strictly increasing singular functions. Furthermore, motivated by the observation that for each famous singular function f investigated in the past, f’(ξ) = 0 if f’(ξ) exists and is finite, we show how, for example, an increasing real function g can be constructed so that $g^{'} (x) = 2^{x}$ for all rational numbers x and g’(x) = 0 for almost all irrational numbers x.

Commutators on ${(\sum ℓ_{q})}_{p}$

Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)

Studia Mathematica

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Let T be a bounded linear operator on $X = {(\sum ℓ_{q})}_{p}$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.