1. If X is a Banach space of type p and of cotype q, then every its n-dimensional subspace is Cn ... more 1. If X is a Banach space of type p and of cotype q, then every its n-dimensional subspace is Cn 1/p−1/q-complemented in X (cf. [1]). Szankowski [2] has showed that if T(X) = sup{p: X of type p} ̸ = 2 or C(X) = inf{q: X of cotype q} ̸ = 2, then X has a subspace without the approximation property. Thus, if each subspace of X possesses the approximation property, then necessarily T(X) = C(X) = 2 and, therefore, all of finite dimensional subspaces in X are ”well ” complemented. Moreover, the space X need not be Hilbertian (or isomorphic to a Hilbert space). The examples of such spaces were constructed by Johnson [3]. In connection with the examples of Szankowski and Johnson, the natural questions arises: 1)if T(X) = C(X) = 2, then is it true that every subspace in X has the approximation property? 2) how ”well complemented ” may be the finite dimensional subspaces of a space without the approximation property: in particular, does there exist a space X without the approximation property...
Abstract. Among other things, it is shown that there exist Banach spaces Z and W such that Z ∗ ∗ ... more Abstract. Among other things, it is shown that there exist Banach spaces Z and W such that Z ∗ ∗ and W have bases, and for every p ∈ [1,2) there is an operator T: W → Z that is not p-nuclear but T ∗ ∗ is p-nuclear. We hold standard notation of the geometrical theory of operators in Banach spaces. The classical reference book on the theory of operator ideals is the A. Pietsch monograph [7]. Notations and terminology we use can be found, for instance, in [7], [8], [10], [11]. For our purposes, it is enough only to recall that if X, Y is a pair of Banach spaces and p> 0, then Np(X, Y) denotes the space of all p-nuclear operators from X to Y, and X ∗̂⊗pY denotes the associated with it p-projective tensor product. And one more important reminder. If J is an operator ideal, then J reg (X, Y) denotes the space of all operators T from X into Y, for which πY T ∈ J(X, Y ∗ ∗), where πY is the canonical isometric imbedding of the space Y into its second dual Y ∗ ∗. In this note we are intere...
The possibility of factoring a product of nuclear operators through operators in the von Neumann–... more The possibility of factoring a product of nuclear operators through operators in the von Neumann–Schatten class is considered. In particular, generally, the product of two nuclear operators can be factored only through a Hilbert–Schmidt operator.
Pour tout p≥1, p¬=2, il existe un espace de Banach separable E qui possede une base de Schauder, ... more Pour tout p≥1, p¬=2, il existe un espace de Banach separable E qui possede une base de Schauder, et un operateur lineaire continu T:E→E tels que T ** soit p-nucleaire, mais que T ne soit pas p-nucleaire
If $ s\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$... more If $ s\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$ to a Banach space $ Y$ and if one of the spaces $ X^*$ or $ Y^{***}$ has the approximation property of order $s,$ $AP_s,$ then the operator $ T$ is nuclear. The result is in a sense exact. For example, it is shown that for each $r\in (2/3, 1]$ there exist a Banach space $Z_0$ and a non-nuclear operator $ T: Z_0^{**}\to Z_0$ so that $Z_0^{**}$ has a Schauder basis, $ Z_0^{***}$ has the $AP_s$ for every $s\in (0,r)$ and $T^*$ is $r$-nuclear.
We give a small survey in connection with the famous Grothendieck's inequality. We consider some ... more We give a small survey in connection with the famous Grothendieck's inequality. We consider some classical applications, an application to the geometry of Banach spaces as well as applications to the well known problem of whether the S p-algebras with their Schur products should be Q-algebras.
The first example of a Banach space with the approximation property but without the bounded appro... more The first example of a Banach space with the approximation property but without the bounded approximation property was given by Figiel and Johnson in 1973. We give the first example of a Banach lattice with the approximation property but without the bounded approximation property. As a consequence, we prove the existence of an integral operator (in the sense of Grothendieck) on a Banach lattice which is not strictly integral.
We study some known approximation properties and introduce and investigate several new approximat... more We study some known approximation properties and introduce and investigate several new approximation properties, closely connected with different quasi-normed tensor products. These are the properties like the $AP_s$ or $AP_{(s,w)}$ for $s\in (0,1],$ which give us the possibility to identify the spaces of $s$-nuclear and $(s,w)$-nuclear operators with the corresponding tensor products (e.g., related to Lorentz sequence spaces). Some applications are given (in particular, we present not difficult proofs of the trace-formulas of Grothendieck-Lidskii type for several ideals of nuclear operators).
It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X... more It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X), for which the closures of convex sets in tau_p and in *-weak topology of the space Pi_p(Y,X) are coincident. Thereafter, we investigate some properties of the space Pi_p, related to this new topology. 2010-remark: Occasionally, the topology is coincides with the lambda_p-topology from the paper "Compact operators which factor through subspaces of l_p", Math. Nachr. 281(2008), 412-423 by Deba Prasad Sinha and Anil Kumar Karn.
1. If X is a Banach space of type p and of cotype q, then every its n-dimensional subspace is Cn ... more 1. If X is a Banach space of type p and of cotype q, then every its n-dimensional subspace is Cn 1/p−1/q-complemented in X (cf. [1]). Szankowski [2] has showed that if T(X) = sup{p: X of type p} ̸ = 2 or C(X) = inf{q: X of cotype q} ̸ = 2, then X has a subspace without the approximation property. Thus, if each subspace of X possesses the approximation property, then necessarily T(X) = C(X) = 2 and, therefore, all of finite dimensional subspaces in X are ”well ” complemented. Moreover, the space X need not be Hilbertian (or isomorphic to a Hilbert space). The examples of such spaces were constructed by Johnson [3]. In connection with the examples of Szankowski and Johnson, the natural questions arises: 1)if T(X) = C(X) = 2, then is it true that every subspace in X has the approximation property? 2) how ”well complemented ” may be the finite dimensional subspaces of a space without the approximation property: in particular, does there exist a space X without the approximation property...
Abstract. Among other things, it is shown that there exist Banach spaces Z and W such that Z ∗ ∗ ... more Abstract. Among other things, it is shown that there exist Banach spaces Z and W such that Z ∗ ∗ and W have bases, and for every p ∈ [1,2) there is an operator T: W → Z that is not p-nuclear but T ∗ ∗ is p-nuclear. We hold standard notation of the geometrical theory of operators in Banach spaces. The classical reference book on the theory of operator ideals is the A. Pietsch monograph [7]. Notations and terminology we use can be found, for instance, in [7], [8], [10], [11]. For our purposes, it is enough only to recall that if X, Y is a pair of Banach spaces and p> 0, then Np(X, Y) denotes the space of all p-nuclear operators from X to Y, and X ∗̂⊗pY denotes the associated with it p-projective tensor product. And one more important reminder. If J is an operator ideal, then J reg (X, Y) denotes the space of all operators T from X into Y, for which πY T ∈ J(X, Y ∗ ∗), where πY is the canonical isometric imbedding of the space Y into its second dual Y ∗ ∗. In this note we are intere...
The possibility of factoring a product of nuclear operators through operators in the von Neumann–... more The possibility of factoring a product of nuclear operators through operators in the von Neumann–Schatten class is considered. In particular, generally, the product of two nuclear operators can be factored only through a Hilbert–Schmidt operator.
Pour tout p≥1, p¬=2, il existe un espace de Banach separable E qui possede une base de Schauder, ... more Pour tout p≥1, p¬=2, il existe un espace de Banach separable E qui possede une base de Schauder, et un operateur lineaire continu T:E→E tels que T ** soit p-nucleaire, mais que T ne soit pas p-nucleaire
If $ s\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$... more If $ s\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$ to a Banach space $ Y$ and if one of the spaces $ X^*$ or $ Y^{***}$ has the approximation property of order $s,$ $AP_s,$ then the operator $ T$ is nuclear. The result is in a sense exact. For example, it is shown that for each $r\in (2/3, 1]$ there exist a Banach space $Z_0$ and a non-nuclear operator $ T: Z_0^{**}\to Z_0$ so that $Z_0^{**}$ has a Schauder basis, $ Z_0^{***}$ has the $AP_s$ for every $s\in (0,r)$ and $T^*$ is $r$-nuclear.
We give a small survey in connection with the famous Grothendieck's inequality. We consider some ... more We give a small survey in connection with the famous Grothendieck's inequality. We consider some classical applications, an application to the geometry of Banach spaces as well as applications to the well known problem of whether the S p-algebras with their Schur products should be Q-algebras.
The first example of a Banach space with the approximation property but without the bounded appro... more The first example of a Banach space with the approximation property but without the bounded approximation property was given by Figiel and Johnson in 1973. We give the first example of a Banach lattice with the approximation property but without the bounded approximation property. As a consequence, we prove the existence of an integral operator (in the sense of Grothendieck) on a Banach lattice which is not strictly integral.
We study some known approximation properties and introduce and investigate several new approximat... more We study some known approximation properties and introduce and investigate several new approximation properties, closely connected with different quasi-normed tensor products. These are the properties like the $AP_s$ or $AP_{(s,w)}$ for $s\in (0,1],$ which give us the possibility to identify the spaces of $s$-nuclear and $(s,w)$-nuclear operators with the corresponding tensor products (e.g., related to Lorentz sequence spaces). Some applications are given (in particular, we present not difficult proofs of the trace-formulas of Grothendieck-Lidskii type for several ideals of nuclear operators).
It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X... more It is a translation of an old paper of mine. We describe the topology tau_p in the space Pi_p(Y,X), for which the closures of convex sets in tau_p and in *-weak topology of the space Pi_p(Y,X) are coincident. Thereafter, we investigate some properties of the space Pi_p, related to this new topology. 2010-remark: Occasionally, the topology is coincides with the lambda_p-topology from the paper "Compact operators which factor through subspaces of l_p", Math. Nachr. 281(2008), 412-423 by Deba Prasad Sinha and Anil Kumar Karn.
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