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In mathematics, Riesz-Schauder theory, named after Frigyes Riesz and Juliusz Schauder, provides a generalisation of the spectral theory of linear operators on finite dimensional complex vector spaces to those operators on (possibly infinite dimensional) complex Banach spaces which are compact.

Overview

Let $X$ be a complex Banach space and $A$ a bounded operator on $X$. If $X$ is finite dimensional, the spectrum of $A$ consists only of eigenvalues of $A$ whose number does not exceed the dimension of $X$. If, however, $X$ is infinite dimensional, the situation becomes much more complex in general. In this case the spectrum may be uncountably infinite, containing not only eigenvalues but also approximate eigenvalues and the compression spectrum of $A$ (see spectrum (functional analysis) for details). Riesz-Schauder theory is concerned with those linear operators $A$ which not only are bounded, but also compact, providing an interstage between these two extremes.

Main theorem

The main theorem of Riesz-Schauder theory is the spectral theorem for compact operators, or simply the Riesz-Schauder theorem. Before the theorem is formally stated, the introduction of certain definitions and notations is advisable.

Given a positive integer $n$ and a complex number $\lambda$, the operator $(\lambda 1-A)^{n}$ will frequently be contemplated. Here, the kernel of this operator will be denoted by $N_{\lambda }^{n}$ while its range will be denoted by $R_{\lambda }^{n}$. The symbol $n_{\lambda }$, called the index of $\lambda$, will denote the largest $n$ such that $N_{\lambda }^{n-1}\neq N_{\lambda }^{n}$ if such an $n$ exists, otherwise $n_{\lambda }$ is assumed to be infinity.

Its formal statement is as follows.

Let $X$ be a complex Banach space and $A$ a compact operator on $X$. Then the following statements hold:

1. The spectrum $S(A)$ of $A$ is countable. With the possible exception of zero it contains only eigenvalues of $A$ and no accumulation points.
2. Given any nonzero eigenvalue $\lambda$ of $A$, the chain $\ker((\lambda 1-A)^{1})\subseteq \ker((\lambda 1-A)^{2})\subseteq \ker((\lambda 1-A)^{3})\subseteq \cdots$ satisfies the ascending chain condition. Moreover, if $n_{\lambda }$ is the greatest integer such that $\ker((\lambda 1-A)^{n-1})\neq \ker((\lambda 1-A)^{n})$

References

• Alt, H. W., Lineare Funktionalanalysis, 3rd ed., Springer, ISBN 3-540-65421-6
• Dunford, N., Schwartz, J. T., Linear Operators Part I, Interscience Publishers, Inc., New York, ISBN 0-470-22605-6