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We consider simultaneous solutions of operator Sylvester equations (1 ≤ i ≤ k), where and are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of and do not intersect, then this system of Sylvester equations has a unique simultaneous solution.
In this paper we suggest a general framework of the spectral mapping theorem in terms of parametrized Banach space bicomplexes.
Let M be a Beurling-type submodule of , the Hardy space over the unit ball of , and let be the associated quotient module. We completely describe the spectrum and essential spectrum of N, and related index theory.
A formula is given for the (joint) spectral radius of an n-tuple of mutually commuting Hilbert space operators analogous to that for one operator. This gives a positive answer to a conjecture raised by J. W. Bunce in [1].
A new concept of spectrum for a solvable Lie algebra of operators is introduced, extending the Taylor spectrum for commuting tuples. This spectrum has the projection property on any Lie subalgebra and, for algebras of compact operators, it may be computed by means of a variant of the classical Ringrose theorem.
We identify how the standard commuting dilation of the maximal commuting piece of any row contraction, especially on a finite-dimensional Hilbert space, is associated to the minimal isometric dilation of the row contraction. Using the concept of standard commuting dilation it is also shown that if liftings of row contractions are on finite-dimensional Hilbert spaces, then there are strong restrictions on properties of the liftings.
We study dilations of q-commuting tuples. Bhat, Bhattacharyya and Dey gave the correspondence between the two standard dilations of commuting tuples and here these results are extended to q-commuting tuples. We are able to do this when the q-coefficients are of modulus one. We introduce a “maximal q-commuting subspace” of an n-tuple of operators and a “standard q-commuting dilation”. Our main result is that the maximal q-commuting subspace of the standard noncommuting dilation of a q-commuting...
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.
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