|
Organizers |
Foundations of the Spectral Theory of Nevanlinna's Mapping from a Topological Space to the Algebra of Endomorphisms of a Banach Space and its Applications
by
M. B. Ragimov
Harran Universty, Sanliurfa
The spectral theory of Nevanlinna's mappings from a topological space Ex to the algebra L(Ey, Ey) of endomorphisms of a Banach space Ey was constructed by the author in [1] and the fundamental spectral theorems proved. Applications to multidimensional differential equations with Nevanlinna's operator-valued coefficients [2] were also given. Various definitions of the spectrum were presented, their coincidences, inclusions and nonemptinesses were proved. The spectral theory and functional calculus for an infinite number of closed commutaing operators was constructed. In particular the following theorem was proved:
Theorem 1 Let A in T(Ex;L(Ey, Ey)) satisfy the
condition \Lambdahk{A(h)A(k)}=0, for allh, k in Ex,
where the symbol \Lambdahk is the Koso symmetrization operator.
Then the spectrum \sigma(A) of the non-zero mapping A in T(Ex;L(Ey, Ey)) is non empty and for any element x in Ex the spectrum
\sigma(A(x)) of the operator A(x) coincides with the set
{\lambda(x)}\lambda in \sigma(A).
We may also mention a coincidence criterion for spectrums for one
special T(Ex;L(Ey, Ey)).
Theorem 2. Let the mapping A : Ex --> L(Ey, Ey)
has the property that each operator A(x), x in Ex, can be presented
in the form A(x)=\alpha(x)IEy + B(x), where \alpha in E*x is
a functional, B(x) is a completely continuous operator and B(h)B(k) = B(k)B(h), for allh, k in Ex. Then the different definitions \sigmai(A)
of the spectrum coincide.
This theory opens new directions in the multidimensional spectral analysis
of linear operators and its applications in theoretical physics, and to
theory of multidimensional differential equations in Banach space.
Consider the multidimensional differential equation:
| (1) |
Definition. The complement of those functionals for which there exists a function f in L&(Ex) such that [^f](\lambda0) =/= 0 and (f * j)(x)=\intExf(x-\sigma)·j(\sigma) d\sigma = 0 is called the Berling spectrum of j, and denoted by S(j).
Then we obtain:
Theorem 3. Let j be a bounded solution of
Equation (1). Then the Berling spectrum of j belongs to the set
\sigma(A) \cap i·E*x.
Here, E*x is the space of functionals defined on the space Ex.
References
Date received: August 3, 2001
Copyright © 2001 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 # cagx-63.