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\begin{document}


Method of left and right pseudoresolvents for construction of spectral
theory of mappings 

acting from one complex Banach space into another

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ M.B.Ragimov

Baku State University, 23, Z.Khalilov St., 370073/1, Baku, Azerbaijan, 

e-mail: mrahimov@harran.edu.tr; shfaig@hotmail.com

In this paper there are considered ordered pairs of linear operators $(A,B)$
from Banach

space $L(X;Y)$ of linear ordered operators, determined on the complex Banach
space $X$ 

with values in complex Banach space $Y$. Essentially attracting the bundle
of operators in

the view $A-\lambda B,\lambda \in C$, the conception of singular set
(spectrum) of ordered pair $(A,B)$ is 

introduced and the functional calculus is constructed. For construction of
spectral theory 

of mappings, acting from one complex \ Banach space into another, there are
attracted

(so called) left and right  pseudoresolvents  of pair $(A,B)$, permitting to
attract the theory

of commutative Banach algebras and spectral  theory of operators, acting in
one space. 

The method of left and right pseudoresolvent was  first given by the author
in works [1], [2].

This method permits to construct spectral theory  of finite and infinite
number of bounded

and unbounded operators, acting in different functional spaces. It is
important to note that, 

as against spectrum of bounded operator (acting in one space), in this case
the spectrum 

of pair $(A,B):$ $X\rightarrow Y$can be, for example, infinite set. Besides
that, it can prove to be empty. 

These phenomena were first discovered by the author in works [1-3] .

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Literature

[1]. M.B. Ragimov. On the new results in theory of linear Bundles of
operators

//Turkish Mathematical journal. IV national Mathematical conferences.
Antakya Hatay. - 1991, p. 97-98.

[2]. M.B. Ragimov. Spectral theory of ordered pairs of linear operators.
Monograph. BGU, Baku-1933.

[3]. M.B. Ragimov. Multidimensional spectral analysis of linear operators.
Doct.dissert. Baku - 1997. 287 c.

\end{document}