Advanced Search

Abstract New linear topological invariants are introduced and utilized to give an isomorphic classification of tensor products of the type $E_{\infty }\left(a\right)\otimes ̂E_{\infty }^{\text{'}}\left(b\right)$, where $E_{\infty }\left(a\right)$ is a power series space of infinite type. These invariants are modifications of those suggested earlier by Zahariuta. In particular, some new results are obtained for spaces of infinitely differentiable functions with values in a locally convex space X. These spaces coincide, up to isomorphism, with spaces L(s’,X) of all continuous linear operators...

For bounded logarithmically convex Reinhardt pairs "compact set - domain" (K,D) we solve positively the problem on simultaneous approximation of such a pair by a pair of special analytic polyhedra, generated by the same polynomial mapping f: D → ℂⁿ, n = dimΩ. This problem is closely connected with the problem of approximation of the pluripotential ω(D,K;z) by pluripotentials with a finite set of isolated logarithmic singularities ([23, 24]). The latter problem has been solved recently for arbitrary...

A complete isomorphic classification is obtained for Köthe spaces $X=K\left(exp[\chi (p-\kappa \left(i\right))-1/p]a_i\right)$ such that $Xq{d}_{\simeq }{X}^2$; here χ is the characteristic function of the interval [0,∞), the function κ: ℕ → ℕ repeats its values infinitely many times, and $a_i\to \infty$. Any of these spaces has the quasi-equivalence property.

New compound geometric invariants are constructed in order to characterize complemented embeddings of Cartesian products of power series spaces. Bessaga's conjecture is proved for the same class of spaces.

The main result is that the existence of an unbounded continuous linear operator T between Köthe spaces λ(A) and λ(C) which factors through a third Köthe space λ(B) causes the existence of an unbounded continuous quasidiagonal operator from λ(A) into λ(C) factoring through λ(B) as a product of two continuous quasidiagonal operators. This fact is a factorized analogue of the Dragilev theorem [3, 6, 7, 2] about the quasidiagonal characterization of the relation (λ(A),λ(B)) ∈ ℬ (which means that all...

For complete Reinhardt pairs “compact set - domain” K ⊂ D in ℂⁿ, we prove Zahariuta’s conjecture about the exact asymptotics $ln{d}_s\left(A_K^D\right)-{((n!s)/\tau (K,D))}^{1/n}$, s → ∞, for the Kolmogorov widths $d_s\left(A_K^D\right)$ of the compact set in C(K) consisting of all analytic functions in D with moduli not exceeding 1 in D, τ(K,D) being the condenser pluricapacity of K with respect to D.

Let ℓ be a Banach sequence space with a monotone norm $\parallel ·{\parallel }_{\ell }$, in which the canonical system $\left(e_i\right)$ is a normalized symmetric basis. We give a complete isomorphic classification of Cartesian products $E_0^{\ell }\left(a\right)\times E_{\infty }^{\ell }\left(b\right)$ where $E_0^{\ell }\left(a\right)=K^{\ell }\left(exp(-p^{-1}{a}_i)\right)$ and $E_{\infty }^{\ell }\left(b\right)=K^{\ell }\left(exp\left(p{a}_i\right)\right)$ are finite and infinite ℓ-power series spaces, respectively. This classification is the generalization of the results by Chalov et al. [Studia Math. 137 (1999)] and Djakov et al. [Michigan Math. J. 43 (1996)] by using the method of compound linear topological invariants developed by the third author....