P-compact Operators Research Articles

On holomorphic mappings with relatively p-compact range

Related to the concept of p-compact operators with p ? [1,?] introduced by Sinha and Karn [20], this paper deals with the space H? Kp (U, F) of all Banach-valued holomorphic mappings on an open subset U of a complex Banach space E whose ranges are relatively p-compact subsets of F. We characterize such holomorphic mappings as those whose Mujica?s linearisations on the canonical predual of H?(U) are p-compact operators. This fact allows us to make a complete study of them. We show that H? Kp is a surjective Banach ideal of bounded holomorphic mappings which is generated by composition with the ideal of p-compact operators and contains the Banach ideal of all right p-nuclear holomorphic mappings. We also characterize holomorphic mappings with relatively p-compact ranges as those bounded holomorphic mappings which factorize through a quotient space of ?p* or as those whose transposes are quasi p-nuclear operators (respectively, factor through a closed subspace of ?p).

Quotient algebras of Banach operator ideals related to non-classical approximation properties

We investigate the quotient algebra AXI:=I(X)/F(X)‾||⋅||I for Banach operator ideals I contained in the ideal of the compact operators, where X is a Banach space that fails the I-approximation property. The main results concern the nilpotent quotient algebras AXQNp and AXSKp for the quasi p-nuclear operators QNp and the Sinha-Karn p-compact operators SKp. The results include the following: (i) if X has cotype 2, then AXQNp={0} for every p≥1; (ii) if X⁎ has cotype 2, then AXSKp={0} for every p≥1; (iii) the exact upper bound of the index of nilpotency of AXQNp and AXSKp for p≠2 is max⁡{2,⌈p/2⌉}, where ⌈p/2⌉ denotes the smallest n∈N such that n≥p/2; (iv) for every p>2 there is a closed subspace X⊂c0 such that both AXQNp and AXSKp contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace X⊂c0 such that the compact-by-approximable algebra AX=K(X)/A(X) contains two incomparable countably infinite chains of nilpotent closed ideals.

The ideal of Lipschitz classical p-compact operators and its injective hull

Abstract We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

Banach Compactness and Banach Nuclear Operators

In this paper, we introduce the notion of (uniformly weakly) Banach-compact sets, (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators which generalize p-compact sets, p-compact operators and p-nuclear operators, respectively. Fundamental properties are investigated. Factorizations and duality theorems are given. Injective and surjective hulls are used to show the spaces of (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators are quasi Banach operator ideals.

The ideal of weakly p-compact operators and its approximation property for Banach spaces

The ideal of weakly p-compact operators and its approximation property for Banach spaces

Lipschitz p-compact mappings

We introduce the notion of Lipschitz p-compact operators. We show that they can be seen as a natural extension of the linear p-compact operators of Sinha and Karn and we transfer some properties of the linear case into the Lipschitz setting. Also, we introduce the notions of Lipschitz-free p-compact operators and Lipschitz locally p-compact operators. We compare all these three notions and show different properties. Finally, we exhibit examples to show that these three notions are different.

Compact-like operators in lattice-normed spaces

A linear operator T between two lattice-normed spaces is said to be p-compact if, for any p-bounded net xα, the net Txα has a p-convergent subnet. p-Compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, AM-compact operators, etc. Similar to M-weakly and L-weakly compact operators, we define p-M-weakly and p-L-weakly compact operators and study some of their properties. We also study up-continuous and up-compact operators between lattice-normed vector lattices.

Uniform factorization for p-compact sets of p-compact linear operators

Obtaining a factorization of p-compact linear operators via universal Banach spaces, and using the lifting property of quotient maps for p-compact sets we prove a factorization result for relatively r-compact subsets of p-compact operators, where r≥2, 1≤p≤r<∞. To apply our results to homogeneous polynomials, in particular, we show that relatively p-compact subsets of a Banach space of p-compact operators are collectively p-compact.

Essential pseudospectra and essential norms of band-dominated operators

An operator A on an lp-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of A in the Calkin algebra determines, for example, the Fredholmness of A, the Fredholm index, the essential spectrum, the essential norm and the so-called essential pseudospectrum of A. This coset can be identified with the collection of all so-called limit operators of A. It is known that this identification preserves invertibility (hence spectra). We now show that it also preserves norms and in particular resolvent norms (hence pseudospectra). In fact we work with a generalization of the ideal of compact operators, so-called P-compact operators, allowing for a more flexible framework that naturally extends to lp-spaces with p∈{1,∞} and/or vector-valued lp-spaces.

Unconditionally p-null sequences and unconditionally p-compact operators

We investigate sequences and operators via the unconditionally $p$-summable sequences. We characterize the unconditionally $p$-null sequences in terms of a certain tensor product and then prove that, for every $1 \leq p < \infty $, a subset of a Banach sp

Compact operators which are defined by l p-spaces

Let 1 ≤ p < ∞ and 1 ≤ r ≤ p*, where p* is the conjugate index of p. We say that a linear operator T from a Banach space X to a Banach space Y is (p, r)-compact if the image of the unit ball T(B X ) is contained in (where (a n ) ∈ Bc0 if r = ∞) for some p-summable sequence (y n ) ∈ l p (Y). This encompasses the notion of p-compact operators which are precisely the (p, p*)-compact operators. We describe the quasi-Banach operator ideal structure of the class of (p, r)-compact operators. Our approach is more direct and easier than those that have been developed so far for the study of the class of p-compact operators.

On p-compact mappings and the p-approximation property

The notion of p-compact sets arises naturally from Grothendieckʼs characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of p-approximation property and p-compact operators (which form an ideal with its ideal norm κp). This paper examines the interaction between the p-approximation property and certain space of holomorphic functions, the p-compact analytic functions. In order to understand these functions we define a p-compact radius of convergence which allows us to give a characterization of the functions in the class. We show that p-compact holomorphic functions behave more like nuclear than compact maps. We use the ϵ-product of Schwartz, to characterize the p-approximation property of a Banach space in terms of p-compact homogeneous polynomials and in terms of p-compact holomorphic functions with range on the space. Finally, we show that p-compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the κp-approximation property. Our approach also allows us to solve several questions posed by Aron, Maestre and Rueda (2010).

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

We study the space of $p$-compact operators, $\mathcal K_p$, using the theory of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated to $/d_p$, the left injective associate of the Chevet&#8211;Saphar tensor norm $d_p$ (which is e

A remark on the approximation of p-compact operators by finite-rank operators

The Banach operator ideal K p of p-compact operators was introduced in [P.D. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ℓ p , Studia Math. 150 (2002) 17–33]. Let p ⩾ 1 and let X be a closed subspace of an L p ( μ ) -space. We show that X has the approximation property if and only if for every Banach space Y, the linear space F ( Y , X ) of finite-rank operators is K p -norm dense in K p ( Y , X ) , i.e., X has the K p -approximation property.

Density of finite rank operators in the Banach space of p-compact operators

A Banach space X is said to have the k p -approximation property ( k p -AP) if for every Banach space Y, the space F ( Y , X ) of finite rank operators is dense in the space K p ( Y , X ) of p-compact operators endowed with its natural ideal norm k p . In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the k p -AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the k p -AP and Saphar's approximation property. As another application, the k p -AP is characterized in terms of a trace condition. Finally, we relate the k p -AP to the ( p , p ) -approximation property introduced in Sinha and Karn (2002) [15] for subspaces of L p ( μ ) -spaces.

Tensor products and Banach ideals of p-compact operators

A study is made of the norm wp (1 ≤ p ≤ ∞) on the tensor product of two Banach spaces E and F. It is shown that wp is a tensor norm, and a representation is deduced for the elements in the completion\(E\tilde \otimes _{w_p } F\) of E ⊗ F equipped with wp. Finally it is shown that the wp-nuclear operators in the sense of Grothendieck [3] coincide with those operators factoring compactly throughp (if 1 ≤ p ≤ ∞) or Co (if p=∞), with related equalities concerning the idea1 norms.

Banach ideals of p-compact operators

The main theme of this paper is a study in some detail of Banach ideals of continuous linear operators between Banach spaces factoring compactly through lp (1≤p<∞) or co, called p-compact and ∞-compact operators respectively. Recently operators of these types have been studied in [4] within the framework of locally convex spaces which are dense subspaces of p-compact projective limits of Banach spaces. These ideals show close resemblance to the ideals of p-nuclear operators-for the case p=∞ they coincide. Analogously to results of Grothendieck concerning continuous linear operators, we consider vector sequence spaces isometric isomorphic to certain spaces of compact linear operators. A representation theorem for p-compact operators is deduced and isometric properties of the ideal norm are treated. The paper also includes some remarks on unconditional convergence and related operator ideals and a representation for the complete ɛ-tensor product\(\ell^{P}\tilde{\bigotimes}_{\varepsilon}E\) (1≤p<∞) is given.

On some existence theorems for the solutions of a class of nonlinear operator equations

Theorems of M. A. Krasnosel'skii are generalized which concern fixed points of completely continuous operators which compress or expand a cone, and a topological principle for fixed points in the case of P-compact operators.

A Leray-Schauder theorem for a class of nonlinear operators

One of the most useful tools for dealing with the theory of nonlinear operators in concrete analytical problems is the Leray-Schauder fixed point theorem [5]. A disadvantage is that it is only available for the completely continuous (that is, compact and continuous) operators, and moreover the proof of the theorem uses the theory of the topological degree of a mapping. However, in a recent publication, Browder [1] has given an elementary proof without recourse to degree theory, tt was observed by the authors that this method of proof is applicable to a wider class of operators, namely those we shall call strongly P-compact. In this note we shall show that for a restricted class of Banach spaces (including, however, all separable Hilbert spaces) the class of strongly P-compact operators is quite extensive, and shall prove a fixed point theorem like that of Leray-Schauder for this class. Throughout this paper X will denote a real Banach space which has the following property: (re)l: There exists a sequence {F.} of finite dimensional subspaces of X, with F.CF,+I for each n, and U F, dense in X, and a corresponding sequence

On the functional equations involving nonlinear generalized 𝑃-compact operators

Introduction. In his recent papers [23], [24], [25] Petryshyn has introduced and studied class of projectionally compact (P-compact) nonlinear mappings of real Banach space with property (-)K into itself. The main result of this study was an elementary and essentially constructive proof of two fixed point theorems for bounded [23], [24] and unbounded [25] P-compact operators which were then used to derive number of results concerning existence and construction of solutions of various classes of nonlinear equations. Since, as was shown in [25], P-compact operators include, among others, completely continuous, quasicompact, and various classes of monotone operators (in Hilbert space), fixed point theorems (for balls about origin) of Schauder [34], Rothe [32], Krasnoselsky [17], Altman [1] and Kaniel [14] as well as certain existence theorems for monotone operators of Zarantonello [41], Kacurowski [13], Minty [21], Browder [2], [3], Dolph and Minty [10] and others were deduced from results on P-compact operators. Further results for P-compact operators or their extensions have been recently obtained by Lees and Schultz [18], De Figueiredo [8], [9] and Tucker [36]. The purpose of this paper is to continue study and to extend results for P-compact operators obtained in [23], [24], [25] to generalized P-compact or Py-compact operators introduced in [28]. One of our objectives is to show that many properties and results which are true for completely continuous operators carry over to generalized P-compact operators. We now outline briefly main results of this paper. In ?1 we introduce and discuss some basic definitions to be used in this paper. In ?2 we discuss further properties of P-compact and generalized P-compact operators. The main result of this section is Theorem 2.4 which asserts that under certain conditions the Frechet derivative A'(x) is P-compact if and only if A(x) is P-compact. In ?3 we discuss certain properties of linear P-compact operators. The main result of this section is characterization Theorem 3.3 which asserts that a symmetric linear mapping A of Hilbert space into itself is P-compact if and only if A = S+ T where S? 0 and T is completely continuous.