Estudiamos la existencia de subespacios hiperinvariantes de operadores desplazamiento bilateral ponderados e invertibles definidos sobre un espacio de Hilbert con base ortogonal {en}, n perteneciendo a Z, por la expresión T en = wn en+1, donde las sucesiones {wn} y {w-n}, con n = 1, ..., ∞, son convergentes.
Let T: C¹(ℝ) → C(ℝ) be an operator satisfying the “chain rule inequality” T(f∘g) ≤ (Tf)∘g⋅Tg, f,g ∈ C¹(ℝ). Imposing a weak continuity and a non-degeneracy condition on T, we determine the form of all maps T satisfying this inequality together with T(-Id)(0) < 0. They have the form Tf = ⎧ , f’ ≥ 0, ⎨ ⎩ , f’ < 0, with p > 0, H ∈ C(ℝ), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable functions...
Two characterizations of the reductivity of a cyclic normal operator in Hilbert space are proved: the equality of the sets of cyclic and *-cyclic vectors, and the equality L²(μ) = P²(μ) for every measure μ equivalent to the scalar-valued spectral measure of the operator. A cyclic subnormal operator is reductive if and only if the first condition is satisfied. Several consequences are also presented.
Xia proves in [9] that a pure subnormal operator S is completely determined by its self-commutator C = S*S - SS*, restricted to the closure M of its range and the operator Λ = (S*|M)*. In [9], [10], [11] he constructs a model for S that involves this two operators and the so-called mosaic, which is a projection-valued function, analytic outside the spectrum of the minimal normal extension of S. He finds all pure subnormals S with rank C = 2. We will give a complete description of pairs of matrices...
This paper concerns pure subnormal operators with finite rank self-commutator, which we call subnormal operators of finite type. We analyze Xia's theory of these operators [21]-[23] and give its alternative exposition. Our exposition is based on the explicit use of a certain algebraic curve in C2, which we call the discriminant curve of a subnormal operator, and the approach of dual analytic similarity models of [26]. We give a complete structure result for subnormal operators of finite type, which...
For an unbounded operator S the question whether its subnormality can be built up from that of every , the restriction of S to a cyclic space generated by f in the domain of S, is analyzed. Though the question at large has been left open some partial results are presented and a possible way to prove it is suggested as well.
We give a necessary and a sufficient condition for a subset of a locally convex Waelbroeck algebra to have a non-void left joint spectrum In particular, for a Lie subalgebra we have if and only if generates in a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.
We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement coincide with those pairs which are isomorphic to a pair of graphs of bounded linear operators between two other Banach spaces. (2) The pairs of subspaces of a Banach space X with a common complement coincide with those pairs for which there exists an involution...