On an inclusion between operator ideals

Manuel A. Fugarolas

Displaying similar documents to “On an inclusion between operator ideals”

$M$ -ideals of compact operators into $ℓ_{p}$

Kamil John, Dirk Werner (2000)

Czechoslovak Mathematical Journal

Similarity:

We show for $2 \leq p < \infty$ and subspaces $X$ of quotients of $L_{p}$ with a $1$ -unconditional finite-dimensional Schauder decomposition that $K (X, ℓ_{p})$ is an $M$ -ideal in $L (X, ℓ_{p})$ .

Strict u-ideals in Banach spaces

Vegard Lima, Åsvald Lima (2009)

Studia Mathematica

Similarity:

We study strict u-ideals in Banach spaces. A Banach space X is a strict u-ideal in its bidual when the canonical decomposition $X * * * = X * \oplus X^{⊥}$ is unconditional. We characterize Banach spaces which are strict u-ideals in their bidual and show that if X is a strict u-ideal in a Banach space Y then X contains c₀. We also show that $ℓ_{\infty}$ is not a u-ideal.

Johnson's projection, Kalton's property (M*), and M-ideals of compact operators

Olav Nygaard, Märt Põldvere (2009)

Studia Mathematica

Similarity:

Let X and Y be Banach spaces. We give a “non-separable” proof of the Kalton-Werner-Lima-Oja theorem that the subspace (X,X) of compact operators forms an M-ideal in the space (X,X) of all continuous linear operators from X to X if and only if X has Kalton’s property (M*) and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe how Johnson’s projection P on (X,Y)* applies to f ∈ (X,Y)* when f is represented via a Borel (with...

On norm closed ideals in $L (ℓ_{p}, ℓ_{q})$

B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky (2007)

Studia Mathematica

Similarity:

It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X = ℓ_{p}$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L (ℓ_{p} \oplus ℓ_{q})$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L (ℓ_{p}, ℓ_{q})$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L (ℓ_{p}, ℓ_{q})$ , including one that has not been studied before. The proofs use various methods...

Operator Lipschitz functions on Banach spaces

Jan Rozendaal, Fedor Sukochev, Anna Tomskova (2016)

Studia Mathematica

Similarity:

Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form $| | f (B) S - {S f (A) | |}_{(X, Y)} \leq c o n s t | | B S - {S A | |}_{(X, Y)}$ for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on $X = ℓ_{p}$ and $Y = ℓ_{q}$ for p < q. We also study the estimate above in the setting of Banach...

Interpolation by elementary operators

Bojan Magajna (1993)

Studia Mathematica

Similarity:

Given two n-tuples $a = (a_{1}, . . ., a_{n})$ and $b = (b_{1}, . . ., b_{n})$ of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that $E a_{j} = b_{j}$ for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in $A^{n}$ .

On the compact approximation property

Vegard Lima, Åsvald Lima, Olav Nygaard (2004)

Studia Mathematica

Similarity:

We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $(S_{γ})$ of compact operators on X such that $s u p_{γ} | | S_{γ} T | | \leq | | T | |$ and $S_{γ} \to I_{X}$ in the strong operator topology. Similar results for dual spaces are also proved.

On quasi $n$ -ideals of commutative rings

Adam Anebri, Najib Mahdou, Emel Aslankarayiğit Uğurlu (2022)

Czechoslovak Mathematical Journal

Similarity:

Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$ -ideals and the class of $(2, n)$ -ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$ -ideal if $\sqrt{I}$ is an $n$ -ideal of $R .$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$ -ideals, the quasi primary ideals, the $(2, n)$ -ideals and the $p r$ -ideals. Moreover, we use the quasi $n$ -ideals to characterize...

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

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

Studia Mathematica

Similarity:

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,...

Monomial ideals with tiny squares and Freiman ideals

Ibrahim Al-Ayyoub, Mehrdad Nasernejad (2021)

Czechoslovak Mathematical Journal

Similarity:

We provide a construction of monomial ideals in $R = K [x, y]$ such that $μ (I^{2}) < μ (I)$ , where $μ$ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring $R$ , we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on $μ (I^{k})$ that generalize...

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

Similarity:

Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X, | | \cdot | |_{X})$ and $(Y, | | \cdot | |_{Y})$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $| | T (1_{A ₙ} f) {| |}_{Y} \to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

Similarity:

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.

On wsq-primary ideals

Emel Aslankarayiğit Uğurlu, El Mehdi Bouba, Ünsal Tekir, Suat Koç (2023)

Czechoslovak Mathematical Journal

Similarity:

We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity and $Q$ a proper ideal of $R$ . The proper ideal $Q$ is said to be a weakly strongly quasi-primary ideal if whenever $0 \neq a b \in Q$ for some $a, b \in R$ , then $a^{2} \in Q$ or $b \in \sqrt{Q} .$ Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero...

Multiple summing operators on $l_{p}$ spaces

Dumitru Popa (2014)

Studia Mathematica

Similarity:

We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T : l_{4 / 3} \times l_{4 / 3} \to l ₂$ such that none of the associated linear operators...

Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces

Piotr Niemiec

Similarity:

An ideal of N-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for N-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $C D D_{N}$ of all unitary equivalence classes of such N-tuples...