The Collatz-Wielandt quotient for pairs of nonnegative operators

Shmuel Friedland

Displaying similar documents to “The Collatz-Wielandt quotient for pairs of nonnegative operators”

Isomorphic properties in spaces of compact operators

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the definition of $p$ -limited completely continuous operators, $1 \leq p < \infty$ . The question of whether a space of operators has the property that every $p$ -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$ -limited completely continuous evaluation operators.

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

On hyponormal operators in Krein spaces

Kevin Esmeral, Osmin Ferrer, Jorge Jalk, Boris Lora Castro (2019)

Archivum Mathematicum

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In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂 = 𝕂^{+} \oplus 𝕂^{-}$ of the Krein space $𝕂$ with $𝕂^{+}$ and $𝕂^{-}$ invariant under $T$ .

On linear maps leaving invariant the copositive/completely positive cones

Sachindranath Jayaraman, Vatsalkumar N. Mer (2024)

Czechoslovak Mathematical Journal

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The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices $𝒮^{n}$ that leave invariant the closed convex cones of copositive and completely positive matrices ( ${COP}_{n}$ and ${CP}_{n}$ ). A description of an invertible linear map on $𝒮^{n}$ such that $L ({CP}_{n}) \subset C P_{n}$ is obtained in terms of semipositive maps over the positive semidefinite cone $𝒮_{+}^{n}$ and the cone of symmetric nonnegative matrices $𝒩_{+}^{n}$ for $n \leq 4$ , with specific calculations for $n = 2$ . Preserver properties of the Lyapunov...

Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

Dario Fasino, Francesco Tudisco (2016)

Czechoslovak Mathematical Journal

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We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$ . The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$ . Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show that the choice where $X$ is the all-ones matrix allows to estimate the signature of the leading eigenvector of $A$ , generalizing previous results on Perron-Frobenius properties of matrices...

Specialization to the tangent cone and Whitney equisingularity

Arturo Giles Flores (2013)

Bulletin de la Société Mathématique de France

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Let $(X, 0)$ be a reduced, equidimensional germ of an analytic singularity with reduced tangent cone $(C_{X, 0}, 0)$ . We prove that the absence of exceptional cones is a necessary and sufficient condition for the smooth part $𝔛^{0}$ of the specialization to the tangent cone $ϕ : 𝔛 \to ℂ$ to satisfy Whitney’s conditions along the parameter axis $Y$ . This result is a first step in generalizing to higher dimensions Lê and Teissier’s result for hypersurfaces of $ℂ^{3}$ which establishes the Whitney equisingularity of $X$ and its tangent...

Effective Hamiltonians and Quantum States

Lawrence C. Evans (2000-2001)

Séminaire Équations aux dérivées partielles

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We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function $u$ solving the eikonal equation aėȧnd a probability measure $σ$ solving a related transport equation. We present some elementary formal identities relating certain quantum states $ψ$ and $u, σ$ . We show also how...

Quantum expanders and geometry of operator spaces

Gilles Pisier (2014)

Journal of the European Mathematical Society

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We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of $M_{N}$ -spaces needed to represent (up to a constant $C > 1$ ) the $M_{N}$ -version of the $n$ -dimensional operator Hilbert space $O H_{n}$ as a direct sum of copies of $M_{N}$ . We show that, when $C$ is close to 1, this multiplicity grows...

On extensions of families of operators

Oleg Lihvoinen (2023)

Commentationes Mathematicae Universitatis Carolinae

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The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of...

Exponentiations over the quantum algebra $U_{q} (s l_{2} (ℂ))$

Sonia L’Innocente, Françoise Point, Carlo Toffalori (2013)

Confluentes Mathematici

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We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra $U_{q} (s l_{2} (ℂ))$ . We discuss two cases, according to whether the parameter $q$ is a root of unity. We show that the universal enveloping algebra of $s l_{2} (ℂ)$ embeds in a non-principal ultraproduct of $U_{q} (s l_{2} (ℂ))$ , where $q$ varies over the primitive roots of unity.

Generalized Cesàro operators on certain function spaces

Sunanda Naik (2010)

Annales Polonici Mathematici

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Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators $^{b, c}$ is bounded on the Dirichlet spaces $_{p, a}$ . We also give a short and direct proof of boundedness of $^{b, c}$ on the Hardy space $H^{p}$ for 1 < p < ∞.

Covariantization of quantized calculi over quantum groups

Seyed Ebrahim Akrami, Shervin Farzi (2020)

Mathematica Bohemica

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We introduce a method for construction of a covariant differential calculus over a Hopf algebra $A$ from a quantized calculus $d a = [D, a]$ , $a \in A$ , where $D$ is a candidate for a Dirac operator for $A$ . We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra $A^{\circ}$ . We apply this method to the Dirac operator for the quantum $SL (2)$ given by S. Majid. We find that the differential calculus obtained by our...

$C^{*}$ -basic construction between non-balanced quantum doubles

Qiaoling Xin, Tianqing Cao (2024)

Czechoslovak Mathematical Journal

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For finite groups $X$ , $G$ and the right $G$ -action on $X$ by group automorphisms, the non-balanced quantum double $D (X; G)$ is defined as the crossed product ${(ℂ X^{op})}^{*} ⋊ ℂ G$ . We firstly prove that $D (X; G)$ is a finite-dimensional Hopf $C^{*}$ -algebra. For any subgroup $H$ of $G$ , $D (X; H)$ can be defined as a Hopf $C^{*}$ -subalgebra of $D (X; G)$ in the natural way. Then there is a conditonal expectation from $D (X; G)$ onto $D (X; H)$ and the index is $[G; H]$ . Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the...

The ideal of p-compact operators: a tensor product approach

Daniel Galicer, Silvia Lassalle, Pablo Turco (2012)

Studia Mathematica

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We study the space of p-compact operators, $_{p}$ , using the theory of tensor norms and operator ideals. We prove that $_{p}$ is associated to $/ d_{p}$ , the left injective associate of the Chevet-Saphar tensor norm $d_{p}$ (which is equal to $g_{p^{'}}^{'}$ ). This allows us to relate the theory of p-summing operators to that of p-compact operators. Using the results known for the former class and appropriate hypotheses on E and F we prove that $_{p} (E; F)$ is equal to $_{q} (E; F)$ for a wide range of values of p and q, and show that our results...