Topological Radicals, II. Applications to Spectral Theory of Multiplication Operators

Abstract

We develop the tensor spectral radius technique and the theory of the tensor radical. Basing on them we obtain several results on spectra of multiplication operators on Banach bimodules and indicate some applications to the spectral theory of elementary and multiplication operators on Banach algebras and modules with various compactness properties. ContentsVICTOR S. SHULMAN AND YURII V. TUROVSKII 6.2. Invariant subspaces for operators on an ordered pair of Banach spaces 51 6.3. Semicompact multiplication operators 52 6.4. Semicompact elementary operators 55 References 61

Topological Radicals, V. From Algebra to Spectral Theory

“…The norm of a family does not depend on the order, so one can consider the corresponding equivalence relation ≃ between families (see details in [ST6]). Define the product M N within up to ≃ as a family H = (c k ) ∞ 1 where c k = a i b j for k = φ(i, j) and φ is an arbitrary bijection of N×N onto N. Then the power M n+1 is defined by M n+1 ≃ M M n for every n > 0 and the tensor (spectral) radius ρ t (M ) is defined by…”

“…. )); then R t (A) is defined as the set of all a ∈ A such that ρ t ({a} ⊔ M ) = ρ t (M ) for every summable family M in A; R t is a uniform topological radical [ST5,ST6].…”

“…The term tensor is justified by the fact that a ∈ R t (A) if and only if a ⊗ b ∈ Rad A ⊗B for every normed algebra B and b ∈ B [ST6,Theorem 3.36]. By [ST6,Theorem 3.29],…”

“…Let (J α ) α≤γ be the increasing transfinite chain of ideals of A constructed in Theorem 2.20(5) with J 0 = 0 and J γ = J. As each J α+1 /J α is bifinite, J α+1 /J α is approximable, so J is closedhypofinite (see [ST6,Proposition 3.48]). As R hf (A) is the largest closed-hypofinite ideal of A then J ⊂ R hf (A).…”

“…The functional analytic counterpart of this theory was initiated by Peter Dixon in his paper [D3], which contained basic definitions and presented first applications. Since then the theory was developed and applied to different problems of operator theory and Banach algebras in [ST2,ST5,ST6,ST7,KST]. The problems that turned out to be "solvable in radicals" had their origin in the theory of invariant subspaces, irreducible representations, semigroups of operators, tensor products, linear operator equations, joint spectral radius, Banach Lie algebras and other topics.…”