On the positivity of semigroups of operators

Roland Lemmert; Peter Volkmann

Displaying similar documents to “On the positivity of semigroups of operators”

Inverses of generators of nonanalytic semigroups

Ralph deLaubenfels (2009)

Studia Mathematica

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Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${e^{t A}}_{t \geq 0}$ . It is shown that $A^{- 1}$ generates an $O (1 + τ) A {(1 - A)}^{- 1}$ -regularized semigroup. Several equivalences for $A^{- 1}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of ${e^{t A}}_{t \geq 0}$ , on subspaces, for $A^{- 1}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate...

Hukuhara's differentiable iteration semigroups of linear set-valued functions

Andrzej Smajdor (2004)

Annales Polonici Mathematici

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Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. A family $F^{t} : t \geq 0$ of continuous linear set-valued functions $F^{t} : K \to c c (K)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $Φ (t, x) = F^{t} (x)$ is a solution of the problem $D_{t} Φ (t, x) = Φ (t, G (x)) : = ⋃ Φ (t, y) : y \in G (x)$ , Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_{t} Φ (t, x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G (x) : = l i m_{s \to 0 +} (F^{s} (x) - x) / s$ for x ∈ K.

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...

Operator theoretic properties of semigroups in terms of their generators

S. Blunck, L. Weis (2001)

Studia Mathematica

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Let $(T_{t})$ be a C₀ semigroup with generator A on a Banach space X and let be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to if and only if the integrated semigroup $S_{t} : = \int_{0}^{t} T_{s} d s$ belongs to . For analytic semigroups, $S_{t} \in$ implies $T_{t} \in$ , and in this case we give precise estimates for the growth of the -norm of $T_{t}$ (e.g. the trace of $T_{t}$ ) in terms of the resolvent growth and the imbedding D(A) ↪ X.

Universal Jamison spaces and Jamison sequences for C₀-semigroups

Vincent Devinck (2013)

Studia Mathematica

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An increasing sequence ${(n_{k})}_{k \geq 0}$ of positive integers is said to be a Jamison sequence if for every separable complex Banach space X and every T ∈ ℬ(X) which is partially power-bounded with respect to ${(n_{k})}_{k \geq 0}$ , the set $σ_{p} (T) \cap$ is at most countable. We prove that for every separable infinite-dimensional complex Banach space X which admits an unconditional Schauder decomposition, and for any sequence ${(n_{k})}_{k \geq 0}$ which is not a Jamison sequence, there exists T ∈ ℬ(X) which is partially power-bounded with respect to ${(n_{k})}_{k \geq 0}$ ...

Semigroups generated by convex combinations of several Feller generators in models of mathematical biology

Adam Bobrowski, Radosław Bogucki (2008)

Studia Mathematica

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Let be a locally compact Hausdorff space. Let $A_{i}$ , i = 0,1,...,N, be generators of Feller semigroups in C₀() with related Feller processes $X_{i} = X_{i} (t), t \geq 0$ and let $α_{i}$ , i = 0,...,N, be non-negative continuous functions on with $\sum_{i = 0}^{N} α_{i} = 1$ . Assume that the closure A of $\sum_{k = 0}^{N} α_{k} A_{k}$ defined on $⋂_{i = 0}^{N} (A_{i})$ generates a Feller semigroup T(t), t ≥ 0 in C₀(). A natural interpretation of a related Feller process X = X(t), t ≥ 0 is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ , with probability...

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

Analytic semigroups on vector valued noncommutative $L^{p}$ -spaces

Cédric Arhancet (2013)

Studia Mathematica

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We give sufficient conditions on an operator space E and on a semigroup of operators on a von Neumann algebra M to obtain a bounded analytic or R-analytic semigroup ( ${(T \otimes I d_{E})}_{t \geq 0}$ on the vector valued noncommutative $L^{p}$ -space $L^{p} (M, E)$ . Moreover, we give applications to the $H^{\infty} (Σ_{θ})$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.

Growth of semigroups in discrete and continuous time

Alexander Gomilko, Hans Zwart, Niels Besseling (2011)

Studia Mathematica

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We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ (t) = A^{- 1} x (t)$ , and the difference equation $x_{d} (n + 1) = (A + I) {(A - I)}^{- 1} x_{d} (n)$ are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup ${(e^{A^{- 1} t})}_{t \geq 0}$ is O(∜t), and for $((A + I) {(A - I)}^{- 1}) ⁿ$ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore,...

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...

On semigroups of $σ$ -endomorphisms

Iwanik, Anzelm

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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.

Spaces of compact operators on $C (2^{} \times [0, α])$ spaces

Elói Medina Galego (2011)

Colloquium Mathematicae

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We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{} \times [0, α]$ , the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

Compact operators whose adjoints factor through subspaces of $l_{p}$

Deba P. Sinha, Anil K. Karn (2002)

Studia Mathematica

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For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $K \subset \sum_{n = 1}^{\infty} α ₙ x ₙ : α ₙ \in B a l l (l_{p^{'}})$ , where p’ = p/(p-1) and $x ₙ \in l_{p}^{s} (X)$ . An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $x ₙ \in l_{p}^{w} (X)$ . It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_{p}$ in a particular manner. The normed operator ideals $(K_{p}, κ_{p})$ of p-compact operators and $(W_{p}, ω_{p})$ of weakly p-compact operators, arising from these factorizations,...

Essentially Incomparable Banach Spaces of Continuous Functions

Rogério Augusto dos Santos Fajardo (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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We construct, under Axiom ♢, a family ${(C (K_{ξ}))}_{ξ < 2^{(2^{ω})}}$ of indecomposable Banach spaces with few operators such that every operator from $C (K_{ξ})$ into $C (K_{η})$ is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size $2^{ω}$ instead of $2^{(2^{ω})}$ .

A local Landau type inequality for semigroup orbits

Gerd Herzog, Peer Christian Kunstmann (2014)

Studia Mathematica

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Given a strongly continuous semigroup ${(S (t))}_{t \geq 0}$ on a Banach space X with generator A and an element f ∈ D(A²) satisfying $| | S (t) f | | \leq e^{- ω t} | | f | |$ and $| | S (t) A ² f | | \leq e^{- ω t} | | A ² f | |$ for all t ≥ 0 and some ω > 0, we derive a Landau type inequality for ||Af|| in terms of ||f|| and ||A²f||. This inequality improves on the usual Landau inequality that holds in the case ω = 0.