Spectral operator

spectral measure

A bounded linear operator $A$ mapping a Banach space $X$ into itself and such that for the $\sigma$- algebra ${\mathcal B}$ of Borel subsets $\delta$ in the plane there is a resolution of the identity $E ( \delta )$ with the following properties: 1) for any $\delta \in {\mathcal B}$ the projector $E ( \delta )$ reduces $A$, that is, $E ( \delta ) A = A E ( \delta )$ and the spectrum $\sigma ( A _ \delta )$ lies in $\overline \delta \;$, where $A _ \delta$ is the restriction of $A$ to the invariant subspace $E ( \delta ) X$; 2) the mapping $\delta \mapsto E ( \delta )$ is a homeomorphism of ${\mathcal B} = \{ \delta \}$ into the Boolean algebra $\{ E ( \delta ) \}$; 3) all projectors $E ( \delta )$ are bounded, that is, $\| E ( \delta ) \| \leq M$, $\delta \in {\mathcal B}$; and 4) the resolution of the identity $E ( \delta )$ is countably additive in the strong topology of $X$, that is, for any $x \in X$ and any sequence $\{ \delta _ {n} \} \subset {\mathcal B}$ of pairwise disjoint sets,

$$E \left ( \cup _ { n= } 1 ^ \infty \delta _ {n} \right ) x = \sum _ { n= } 1 ^ \infty E ( \delta _ {n} ) x .$$

The concept of a spectral operator can be generalized to the case of closed unbounded operators. In 1), the additional requirement is then that the inclusion $E ( \delta ) D ( A) \subset D ( A)$ holds, where $D ( A)$ is the domain of definition of $A$, and $E ( \delta ) X \subset D ( A)$ for bounded $\delta$.

All linear operators on a finite-dimensional space and all self-adjoint and normal operators on a Hilbert space are spectral operators. For example, the operator

$$A x ( t) = t x( t) + \int\limits _ {- \infty } ^ \infty K ( t , s ) x ( s) d s$$

on $L _ {p} ( - \infty , \infty )$, $1 < p < \infty$, is spectral on

$$D ( A) = \left \{ {x ( t) } : { \int\limits _ {- \infty } ^ \infty | t x ( t) | ^ {2} d t < \infty } \right \}$$

if the kernel $K ( t , s )$ is the Fourier transform of a Borel measure $\mu$ on the plane of total variation $\mathop{\rm var} \mu < 1 / 2 \pi$ and is such that

$$\int\limits _ {- \infty } ^ \infty K ( t , s ) x ( s) d s ,\ \int\limits _ {- \infty } ^ \infty K ( t , s ) x ( t) d t$$

are bounded linear operators on $L _ {p} ( - \infty , \infty )$.

Spectral operators have many important properties, such as:

$$\lambda \in \delta ( A) \iff \exists \ \{ x _ {n} \} \subset X , \| x _ {n} \| = 1 , ( A - \lambda I ) x _ {n} \rightarrow 0 .$$

If $X$ is separable, the point and residual spectra of $A$ are at most countable.

References

[1]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , 3 , Interscience (1971)
[2]	N. Dunford, "A survey of the theory of spectral operators" Bull. Amer. Math. Soc. , 64 (1958) pp. 217–274