Some inequalities are proved between the geometric joint spectral radius (cf. [3]) and the joint spectral radius as defined in [7] of finite commuting families of Banach algebra elements.

For a complex nilpotent finite-dimensional Lie algebra of matrices, and a Jordan-Hölder basis of it, we prove a spectral radius formula which extends a well-known result for commuting matrices.

For a commuting n-tuple of matrices we introduce the notion of a joint spectral radius with respect to the p-norm and prove a spectral radius formula.

An operator in a Banach space is called upper (lower) semi-Browder if it is upper (lower) semi-Fredholm and has a finite ascent (descent). We extend this notion to n-tuples of commuting operators and show that this notion defines a joint spectrum. Further we study relations between semi-Browder and (essentially) semiregular operators.

We give a Martinelli-Vasilescu type formula for the Taylor functional calculus and a simple proof of its basic properties.

We study analytic models of operators of class $C_{·0}$ with natural positivity assumptions. In particular, we prove that for an m-hypercontraction $T\in C_{·0}$ on a Hilbert space , there exist Hilbert spaces and ⁎ and a partially isometric multiplier θ ∈ ℳ (H²(),A²ₘ(⁎)) such that ${\cong }_{\theta }=A²ₘ\left(⁎\right)\ominus \theta H²\left(\right)$ and $T\cong P_{{}_{\theta }}{M}_z{|}_{{}_{\theta }}$, where A²ₘ(⁎) is the ⁎-valued weighted Bergman space and H²() is the -valued Hardy space over the unit disc . We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their applications...

We propose a noncommutative holomorphic functional calculus on absolutely convex domains for a Banach algebra homomorphism π of a finite-dimensional solvable Lie algebra 𝔤 in terms of quasispectra σ(π). In particular, we prove that the joint spectral radius of a compact subset in a solvable operator Lie subalgebra coincides with the geometric spectral radius with respect to a quasispectrum.

We consider simultaneous solutions of operator Sylvester equations $A_{i}X-X{B}_i=C_i$ (1 ≤ i ≤ k), where $(A₁,...,A_k)$ and $(B₁,...,B_k)$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C₁,...,C_k)$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A₁,...,A_k)$ and $(B₁,...,B_k)$ 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 $H²{(}_d)$, the Hardy space over the unit ball ${}_d$ of ${ℂ}^d$, and let $N=H²{(}_d)/M$ 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 $q_{ij}$ 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 $S$ of a locally convex Waelbroeck algebra $𝒜$ to have a non-void left joint spectrum ${\sigma }_l\left(S\right).$ In particular, for a Lie subalgebra $L\subset 𝒜$ we have ${\sigma }_l\left(L\right)\ne \varnothing$ if and only if $[L,L]$ generates in $𝒜$ 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.

2000 Mathematics Subject Classification: 47A10, 47A13.In this paper, we give a description of Taylor spectrum of commuting 2-contractions in terms of characteritic functions of such contractions. The case of a single contraction obtained by B. Sz. Nagy and C. Foias is generalied in this work.