Search

We investigate norms of vectors and operators obtained from the infinite scalar matrix where thereby improving and complementing results of [5]. For any choice of and , U gives rise to a bounded linear operator which enjoys compactness properties close to nuclearity. Whereas we cannot characterize for which (p, q) the operator is nuclear, we will show that frequently this is the case, and that for certain Banach operator ideals, the corresponding ideal norms for and/or related operators coincide with the usual operator norm. On the other hand for example, is easily seen to be a Hilbert-Schmidt operator, but neither its norm nor its Hilbert-Schmidt norm are explicitly known. However, at least for the Hilbert-Schmidt norm, reasonably efficient approximation schemes are available.

The question which led to the title of this note is the following: {\it If $X$ is a Banach space and $K$ is a compact subset of $X$, is it possible to find a compact, or even approximable, operator $v:X\to X$ such that $K\subset\ol{v(B_X)}$?} This question was first posed by P.G.Dixon [6] in connection with investigating the problem of the existence of approximate identities in certain operator algebras. We shall provide a couple of observations related to the above question and give in particular a negative answer in case of approximable operators. We shall also provide the first examples of Banach spaces having the approximation property but failing the bounded compact approximation property though all of their duals do even have the metric compact approximation property.

This is a report on a number of recent results on composition operators which map, for 0 < p ⪯ q ∞, the Hardy space Hp (on the unit disk in the complex plane) into H q. Attention is focused on questions of boundedness (existence), compactness, order boundedness and, in connection with the latter, on relating the absolutely summing and nuclearity character as well as special factorization properties of the operator to function theoretic properties of the defining symbol. Moreover, tools are provided to show that certain classes of operators can well be distinguished already on the level of composition operators.

We investigate norms related to the vectors h n =(h n (k)) k ∈ℕ in ℝℕ (or ℂℕ) (n∈ℕ) where $$h_n (k) = \frac{{n!}} {{k\left( {k + 1} \right) \cdots (k + n)}} = \frac{1} {{k\left( {\mathop {n + k}\limits_n } \right)}}. $$ We estimate, and in a few cases even calculate, the norms of the h n ’s as elements of the usual sequence spaces l r . We also show that for ‘almost all’ p, q the matrix \( \mathcal{H} \) with entries h n (k) defines a bounded linear operator from l p into l q , with rather strong compactness properties.

We consider compactness, weak compactness and complete continuity for multiplication operators on von Neumann algebras and their preduals.

This chapter explains the theory of operator ideals that was born in 1941 with the observation of Calkin. Schatten and von Neumann actually dealt with more general classes of operators; the so-called cross-spaces of linear transformations. They phrased their results in the language of direct (now tensor) products of the Banach spaces. To stress the analogy with operator ideals, Gothic bold lowercase letters will be used to denote sequence ideals. The local theory of Banach spaces deals with those properties that can be expressed in terms of finite dimensional spaces. The most usual way is to consider the quantitative relations that are assumed to hold uniformly for any choice of n elements and/or functionals. Grothendieck's theorem has many applications in Banach space geometry. Determining precisely when a concrete operator of prescribed form belongs to a given ideal often provides considerable enlightenment about the operator, its domain and codomain, and the ideal.

In this paper we show that, for analytic composition operators between weighted Bergman spaces (including Hardy spaces) and as far as boundedness, compactness, order boundedness and certain summing properties of the adjoint are concerned, it is possible to modify domain spaces in a systematic fashion: there is a space of analytic functions which embeds continuously into each of the spaces under consideration and on which the above properties of the operator are decided.A remarkable consequence is that, in the setting of composition operators between weighted Bergman spaces, the properties in question can be identified as properties of the operator as a map between appropriately chosen Hilbert spaces.

Many fundamental processes in analysis are best understood by studying and comparing the summability of series in various modes of convergence. This text provides the beginning graduate student, one with basic knowledge of real and functional analysis, with an account of p-summing and related operators. The account is panoramic, with detailed expositions of the core results and highly non-trivial applications to, for example, harmonic analysis, probability and measure theory, and operator theory. Graduate students and researchers from real, complex and functional analysis, and probability theory will benefit from this account.

Publisher Summary This chapter describes some factorization properties of composition operators. The central topic in the study of composition operators on Hardy spaces (as well as on various other function spaces) is the investigation of the interplay between properties of the generating function φ and properties of the operator C φ on the spaces under consideration. β-order bounded composition operators can be characterized by means of a factorization scheme in which one of the factors is a Hilbert–Schmidt operator.

Motivated by the known characterizations of equicontinuity in the dual of a Schwartz space, a nuclear space, or a strongly nuclear space,we introduce the concepts of a Λ-sequence and of a (Λ)-sequence in the dual of an arbitrary lcs [E,τ], and we investigate the corresponding topologies τΛ and τ(Λ) on E of uniform convergence on these sequences. Here Λ is a normal sequence space such that ϕ⊂Λ⊂l∞. Under favorable enough conditions on Λ, including the nuclearity of its normal topology π, [Λ,π] acts as a universal generator for those lcs [E,τ] which satisfy τ=τΛ. Under somewhat weaker assumptions on Λ, [Λ,τ(Λ)] is a universal generator for the lcs [E,τ] with τ=τ(Λ). These results cover e.g. the cases of Λ-nuclear spaces and of nuclear spaces known from the recent literature. As an application we show that every non-trivial ultrabornological lcs is representable as an inductive limit of isomorphic copies of [Λ′, β(Λ ′, Λ)], where Λ is any nuclear power series space of infinite type with stable exponent sequence.

After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz space (l ∞ ,τ(l ∞ ,l 1)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indE a, a∈A, ofseparable Banach spacesE a, and every Mackey null sequence inE is a null sequence in someE a. IfE is ultrabornological, thenE can be represented as lim indE a,a∈A, allE a separable Banach spaces, such that every fast null sequence inE is a null sequence in someE a.