Abstract
The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate’s trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators.
Highlights
The notion of finite potent endomorphism on an arbitrary vector space was introduced by Tate [19] as a basic tool for his elegant definition of Abstract Residues.During the last decade the theory of finite potent endomorphisms have been applied to studying different topics related to Algebra, Arithmetic and Algebraic Geometry
The aim of this work is to study the main properties of bounded finite potent endomorphisms on arbitrary Hilbert spaces
For every bounded finite potent endomorphism we show that Tate’s trace coincides with the Leray trace and with the trace defined by R
Summary
The notion of finite potent endomorphism on an arbitrary vector space was introduced by Tate [19] as a basic tool for his elegant definition of Abstract Residues. The aim of this work is to study the main properties of bounded finite potent endomorphisms on arbitrary Hilbert spaces. For these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. We relate the determinant of a finite potent endomorphism offered in [8] with classical determinants defined with techniques of Functional Analysis for trace class operators. The characterization of these operators is given in Theorem 3.7, the Invariant Subspace Problem is solved for them in Proposition 3.8 and Theorem 3.20 shows that every bounded finite potent endomorphism on a Hilbert space is a Riesz trace class operator. We hope that from the general properties of bounded finite potent endomorphisms introduced in this work, different applications can be found in the near future
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