Noncommutative spectral mapping theorem
by
Anar Dosiev
Institute of Mathematics and Mechanics, Academy of Sciences ofAzerbaijan

Let E be a finite-dimensional Lie algebra embedded into the algebra B( X)  of bounded linear operators on a complex Banach space X. Some generalizations of Taylor and Slodkowski spectra for a family of operators a=( a₁, ... , a_n) generated E, were considered in [1]-[3], when E is solvable Lie algebra. The spectral mapping theorem \sigma( p( a) ) = p(\sigma( a) ) for Taylor spectrum \sigma with respect to noncommuting polynomials p=( p₁, ... , p_m) , was obtained by A.S. Fainshtein in [1] when E is nilpotent Lie algebra and p generates a finite-dimensional Lie subalgebra in the enveloping algebra U( E)  of E.

Our main result is the spectral mapping theorem \sigma_{\delta, k}(f( a) ) = f( \sigma_{\delta, k}( a)) , \sigma_{\pi, k}( f( a) ) = f(\sigma_{\pi, k}( a) ) , for Slodkowski spectra \sigma_{\delta, k}, \sigma_{\pi, k}, with respect to limits of noncommutative ''rational functions'' f of variables a generated nilpotent Lie algebra E.

Let \alpha:E --> B( X) be a representation (i.e. ( X, \alpha) be a Banach E-module) and let A be a topological algebra contained E, such that the closed full subalgebra generated by E is dense in A. Assume that \alpha is extended for a continuous homomorphism of algebras [(\alpha)\tilde]:A --> B(X) (i.e. X is Banach A-module), such that the Koszul complex generated by E-module ( X, \alpha) is a Banach complex of A-modules. In this case the algebra A is said to dominates the E-module ( X, \alpha) . For example of such algebras we introduce a Frechet algebra O_e( r) of holomorphic functions on the open polydisk at the center in zero and with multiradius r, depending on the base e=( e₁, ... , e_n) of the nilpotent Lie algebra E.

For a Lie subalgebra L in A, let L_\alpha=[(\alpha)\tilde](L) , A_\alpha be the closure of [(\alpha)\tilde](A) in B( X) and let \sigma( \alpha(E) ) be one of Slodkowski spectra of the Lie algebra \alpha( E) . Functionals from \sigma( \alpha(E) ) are extended to multiplicative linear functionals of A_\alpha: for any \lambda in \sigma( \alpha( E)) we write \lambda|^A_\alpha in A_\alpha ^* instead of extension of \lambda up to A_\alpha and let \mu|_L\alpha be the restriction of \mu in A_\alpha ^* on L_\alpha, where A_\alpha ^* is the dual space. Assume that the image of L on each member of the Koszul complex is finite-dimensional. For instance, this condition is satisfied if L is finite-dimensional. Then our spectral mapping theorem can be written as the form

\sigma( L\alpha) = ( \sigma( \alpha(E) ) |A\alpha) |L\alpha.

In general case, L_\alpha is solvable Lie (not nilpotent) algebra. Spectra \sigma( L_\alpha) are defined by using a Cartan subalgebras as done in [3].

If we use Lie generators a of \alpha( E) and f of L, then the last formula is reduced to the well known classical spectral mapping formula \sigma( f( a) ) = f( \sigma( a)) .

[1] Fainshtein A.S., Taylor joint spectrum for families of operators generating nilpotent Lie algebra, J. Operator Theory 29 (1993), 3-27.

[2] Boasso E., Dual properties and joint spectra for solvable Lie algebras of operators, J.Operator Theory 33 (1995), 105-116.

[3] Beltita D., Spectrum for a solvable Lie algebra of operators, Stud.Math. 135 (1999), 163-178.

Date received: April 17, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-17.