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