Spectrum and Functions of Operators on Direct Families of Banach Spaces

M. I. Gil
2009 Methods and Applications of Analysis  
We investigate the spectrum, resolvent and analytic functions of bounded linear operators on so called direct families of Banach spaces. The direct families of Banach spaces can be considered on the one hand as a particular generalization of the notion of the direct integral of Hilbert spaces introduced by von Neumann, and on the other hand as a generalization of Banach spaces with mixed norms. As it is shown in the paper, the direct families of Banach spaces is a natural tool for the
more » ... l for the investigation of partial integral operators of the type where w(., .) and K(., ., .) and φ are given functions. Partial integral operators play an essential role in numerous applications, in particular, in physics, mechanics, kinetic and transport theories, etc. We also discuss applications of our results to the Barbashin type integro-differential equations.
doi:10.4310/maa.2009.v16.n4.a7 fatcat:v66wzov5snf6dgynv7l25ydgda