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In this paper we study the entropy numbers of composition operators \(S=TD\) where \(D\) is a diagonal operator generated by a sequence belonging to some generalized Lorentz sequence space and \(T\) is a linear bounded operator with image in a Banach space \(Y\). We highlight the special case of this setting where \(Y\) is a Banach space of type \(p\). Results can be used to obtain entropy estimates of absolutely convex hulls in Banach spaces of type \(p\).

In the paper we continue the investigations of the papers (Carl in Proc. Am. Math. Soc. 137:155–159, 2009, and Carl and Hinrichs in Bull. Lond. Math. Soc. 41:332–340, 2009) between s-numbers and eigenvalues of operators in Banach spaces. We introduce new s-numbers as well as quasi s-numbers generated by the classical s-numbers. An elementary mixing multiplicativity observed in (Carl in Proc. Am. Math. Soc. 137:155–159, 2009) will be very useful for establishing Weyl inequalities for operators in Banach spaces.

Let s = (sn) be an injective s-number sequence in the sense of Pietsch. We show the following Weyl inequality between geometric means of eigenvalues and snumbers for a Riesz-operator T : X ! X acting on a (complex) Banach space of weak type 2 : For any 0 < ± · 1 and all n 2 IN, A n Y i=1 |¸i(T)| ! 1 n

Let s = (sn) be an injective and surjective s-number sequence in the sense of Pietsch. We show for a Riesz-operator T: X -X acting on a (complex) Banach space the following Weyl inequality between geometric means of eigenvalues and s-numbers: For any 0 1 is an absolute constant. The proof rests on an elementary mixing multiplicativity of an arbitrary s-number sequence and a striking result of G. Pisier. The inequality is a contribution to the problem of estimating eigenvalues by s-numbers first started in a strong sense by H. K6nig (1986, 2001).

Let (s n ) be an s-number sequence. We show for each k = 1, 2, . . . and n ≥ k + 1 the inequality between the eigenvalues and s-numbers of a compact operator T in a Banach space. Furthermore, the constant (k + 1)1/2 is optimal for n = k + 1 and k = 1, 2, . . .. This inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand we prove that the Weyl numbers form a minimal multiplicative s-number sequence and by a well-known inequality between eigenvalues and Weyl numbers due to A. Pietsch they are very good quantities for investigating the optimal asymptotic behavior of eigenvalues.

A celebrated result of Johnson, Maurey, Konig and Retherford from 1977 states that for \(2 \leqslant p < \infty \) every complex \(n \times n\) matrix \(T = (\tau _{ij} )_{i,j} \) satisfies the following eigenvalue estimate: $$\left( {\sum\limits_{i = 1}^n {\left| {\lambda _i \left( T \right)} \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \leqslant \left( {\sum\limits_{j = 1}^n {\left( {\sum\limits_{i = 1}^n {\left| {\tau _{ij} } \right|^{p'} } } \right)} {p \mathord{\left/ {\vphantom {p {p'}}} \right. \kern-\nulldelimiterspace} {p'}}} \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \cdot $$ Based on the concept of entropy numbers and a simple product trick we give a selfcontained elementary proof.

We give diverse coverings of convex sets and sections of convex sets in R n by balls and cylinder sets, and establish volume inequalities which generalize the famous Santalo and inverse Santalo inequalities. The tools which we use are developed from the viewpoint of geometric and functional analysis.

The C0-semigroup method is used to find exact solutions of the Korteweg-de Vries (KdV) equation ut = uxxx + 6uux. Various new kinds of solutions such as multisoliton solutions with unbounded phase numbers, “null” solutions and solutions of “wavelet-type” are found. The realization program consists of two steps. Firstly, the scalar KdV equation will be translated into an operator equation Ut = Uxxx + 3(UU)x on the Banach algebra of a given Banach space. The solution formula for the operator equation has been constructed (Proposition 2.1). Secondly, once all of the operators U(x, t) are of rank one, solutions of the scalar KdV equation will be extracted by using the trace functional. The use of fractional powers of a closed unbounded linear operator in a Banach space also plays an essential role in the present work.

An operator-theoretic method for the investigation of nonlinear equations in soliton physics is discussed comprehensively. Originating from pioneering work of Marchenko, our operator-method is based on new insights into the theory of traces and determinants on operator ideals. Therefore, we give a systematic and concise approach to some recent developments in this direction which are important in the context of this paper. Our method is widely applicable. We carry out the corresponding arguments in detail for the Kadomtsev-Petviashvili equation and summarize the results concerning the Korteweg-de Vries and the modified Korteweg-de Vries equation as well as for the sine-Gordon equation. Exactly the same formalism works in the discrete case, as the treatment of the Toda lattice, the Langmuir and the Wadati lattice shows. AMS classification scheme numbers: 35C05, 35Q51, 35Q53, 35Q58, 47D50, 47N20

LetEnandFmbe two symmetric Banach spaces, andαa reasonable norm on their tensor product. We give asymptotically best possible estimates for the approximation and Gelfand numbers of the natural embedding from thenm-dimensional Hilbert spacelnm2intoEn⊗αFm, and its inverse. Our results are used in order to compute some related characteristics of such tensor products (e.g., type and cotype constants).

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets \({K\subset H}\) of a Hilbert space H by the metric entropy of the set K where the covering numbers \({N(K, \varepsilon)}\) of K by \({\varepsilon}\) -balls of H satisfy the Lorentz condition $$ \int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty $$ for some fixed \({0 < r, s \le \infty}\) with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, −∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors.

In recent time much attention has been devoted to the study of entropy of convex hulls in Hilbert and Banach spaces and their applications in different branches of mathematics. In this paper we show how the rate of decay of the dyadic entropy numbers of a precompact set A of a Banach space X of type p , 1 p ≤ 2 , reflects the rate of decay of the dyadic entropy numbers of the absolutely convex hull aco ( A ) of A . Our paper is a continuation of the paper (Carl et al., 2013), where this problem has been studied in the Hilbert space case. We establish optimal estimates of the dyadic entropy numbers of aco ( A ) in the non-critical cases where the covering numbers N ( A , e ) of A by e -balls of X satisfy the Lorentz condition ∫ 0 ∞ ( log 2 ( N ( A , e ) ) ) s / r d e s ∞ for 0 r p ′ , 0 s ∞ or ∫ 0 ∞ ( log 2 ( 2 + log 2 ( N ( A , e ) ) ) ) − α s ( log 2 ( N ( A , e ) ) ) s / r d e s ∞ for p ′ r ∞ , 0 s ≤ ∞ and α ∈ R , with the usual modifications in the case s = ∞ . The integral here is an improper Stieltjes integral and p ′ is given by the Holder condition 1 / p + 1 / p ′ = 1 . It turns out that, for fixed s , the entropy of the absolutely convex hull drastically changes if the parameter r crosses the point r = p ′ . It is still an open problem what happens if r = p ′ and 0 s ∞ . However, in the case s = ∞ we consider also the critical case r = p ′ and, especially, the Hilbert space case r = 2 . We use the results for estimating entropy and Kolmogorov numbers of diverse operators acting from a Banach space whose dual space is of type p or, especially, from a Hilbert space into a C ( M ) space. In particular, we get entropy estimates of operators factoring through a diagonal operator and of abstract integral operators as well as of weakly singular convolution operators. Moreover, estimates of entropy and Kolmogorov numbers of the classical and generalized Riemann–Liouville operator are established, complementing and extending results in the literature.

Several results in the theory of ( p , q ) (p,q) -summing operators are improved by a unified but elementary tensor product concept.

We show, for any operatorT from aC(K)-space into a Banach space with rank (T)≤n, the inequality $$\pi _p (T) \leqslant C(1 + log n)^{1 - 1/p} \pi _{p,1} (T), 1< p< \infty $$ , whereC≤4.671 is a numerical constant. The factor (1+logn)1−1/p is asymptotically correct. This inequality extends a result of Jameson top ≠ 2. Several applications are given — one is a positive solution of a conjecture of Rosenthal and Szarek: For 1≤p

In this paper we investigate the problem how the metric entropy of the image Im(K) of a bounded ‘abstract kernel’ K: X → E′ mapping an arbitrary set X into the dual E′ of a Banach space E reflects the rate of decay of approximation quantities of the induced operator $$ ({T_K}x)(s): = {\text{for}}x \in E{\text{ and }}s \in X, $$ considered from E into l ∞(X). In the case of Hilbert spaces, we give sufficient and optimal conditions for the metric entropy of Im(K) which guarantee that the induced integral operator T K : L 2 (Y, v) → L 2(X, μ), where (X, μ), (Y, v) are finite measure spaces, belongs to the Schatten classes S q,t . In order to illustrate the usefulness of our results we apply them to eigenvalue problems.

Entropy quantities are connected with the 'degree of compactness' of compact or precompact spaces, and so are appropriate tools for investigating linear and compact operators between Banach spaces. The main intention of this Tract is to study the relations between compactness and other analytical properties, e.g. approximability and eigenvalue sequences, of such operators. The authors present many generalized results, some of which have not appeared in the literature before. In the final chapter, the authors demonstrate that, to a certain extent, the geometry of Banach spaces can also be developed on the basis of operator theory. All mathematicians working in functional analysis and operator theory will welcome this work as a reference or for advanced graduate courses.

In the paper we prove two inequalities involving Gelfand numbers of operators with values in a Hilbert space. The first inequality is a Rademacher version of the main result in [Pa-To-1] which relates the Gelfand numbers of an operator from a Banach spaceX intol 2 with a certain Rademacher average for the dual operator. The second inequality states that the Gelfand numbers of an operatoru froml 1 into a Hilbert space satisfy the inequality $$k^{1/2} c_k (u) \leqq C\parallel u\parallel (\log (1 + N/k))^{1/2}$$ whereC is a universal constant. Several applications of these inequalities in the geometry of Banach spaces are given.

In the paper local entropy moduli of operators between Banach spaces are introduced. They constitue a generalization of entropy numbers and moduli, and localize these notions in an appropriate way. Many results regarding entropy numbers and moduli can be carried over to local entropy moduli. We investigate relations between local entropy moduli and s-numbers, spectral properties, eigenvalues, absolutely summing operators. As applications, local entropy moduli of identical and diagonal operators between lp-spaces can be estimated. It is shown, that in general local and global degree of compactness considerably differ, but under certain type assumptions on the underlying Banach spaces they coincide. Finally, the results are applied to obtain (optimal) estimates for eigenvalues of certain integral operators.

Relations concerning entropy quantities and s-numbers are established. In particular, we derive inequalities between entropy numbers and Weyl numbers. Furthermore, we give eigenvalue distributions for operators of lp,u-type.

We determine the asymptotic behaviour of Kolmogorov and approximation numbers of integral operators, which act in C ( X ), where X is a compact metric space, and which are defined by means of Holder continuous kernels. The results show the connection of the decay of these s -numbers with the entropy (and in this way with the dimension) of X .

In this paper we characterize diagonal operators from ZP into l,, 1

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

On caracterise le degre de compacite des operateurs absolument sommants agissant entre des espaces de Banach particuliers en fonction des nombres d'entropie

We establish inequalities between entropy numbers and approximation numbers for operators acting between Banach spaces. Furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a Banach space. The results are compared with the classical inequalities of Bernstein and Jackson.

Let S be an operator admitting a factorization where D is a diagonal operator and T an (arbitrary) operator with the image in a Banach space of type p . We shall characterize these operators by entropy numbers. Using a well-known inequality between eigenvalues and entropy numbers we establish some results about distributions of eigenvalues.