Spectrum of an operator

$A$

The set $\sigma ( A)$ of complex numbers $\lambda \in \mathbf C$ for which the operator $A- \lambda I$ does not have an everywhere-defined bounded inverse. Here, $A$ is a linear operator on a complex Banach space $X$ and $I$ is the identity operator on $X$. If $A$ is not closed on $X$, then $\sigma ( A) = \mathbf C$, and therefore one usually considers spectra of closed operators (the spectrum of the closure of an operator for operators admitting a closure is sometimes called the closure spectrum).

If $A- \lambda I$ is either non-injective or non-surjective, then $\lambda \in \sigma ( A)$. In the first case $\lambda$ is called an eigenvalue of $A$; the set $\sigma _ {p} ( A)$ of eigenvalues is called the point spectrum. In the second case $\lambda$ is called a point of the continuous spectrum $\sigma _ {c} ( A)$ or the residual spectrum $\sigma _ {r} ( A)$, depending on whether the subspace $( A- \lambda I) X$ is dense in $X$ or not.

There are also other classifications of the points of a spectrum. For example, $\sigma ( A)= \sigma _ {a} ( A) \cup \sigma _ {d} ( A)$, where $\sigma _ {a} ( A)$ consists of approximate eigenvalues ( $\lambda \in \sigma _ {a} ( A)$ if there are $\{ x _ {n} \} \subset X$ with $\| x _ {n} \| = 1$ such that $\| ( A- \lambda I) x _ {n} \| \rightarrow 0$), and

$$\sigma _ {d} ( A) =$$

$$= \ \{ {\lambda \in \mathbf C } : { \mathop{\rm Ker} ( A- \lambda I) = 0 , \overline{ {( A- \lambda I ) X }}\; = ( A - \lambda I ) X \neq X } \} .$$

Note that $\sigma _ {d} ( A) \subset \sigma _ {r} ( A)$, and so $\sigma _ {p} ( A) \cup \sigma _ {c} ( A) \subset \sigma _ {a} ( A)$. In perturbation theory, use is made of the limit spectrum $\sigma _ {\lim\limits} ( A)$, which consists of the limit points of $\sigma ( A)$ and the isolated eigenvalues of infinite multiplicity, of the Weyl spectrum, which is equal to the intersection of the spectra of all compact perturbations, etc.

If $A$ is a bounded operator, then $\sigma ( A)$ is compact and non-empty (in this case $\sigma ( A)$ coincides with the spectrum of the element $A$ of the Banach algebra $B( X)$, cf. Spectrum of an element); in general one can only say that $\sigma ( A)$ is closed in $\mathbf C$. On the set $\rho ( A) = \mathbf C \setminus \sigma ( A)$ one can define the analytic $B( X)$- valued function $R _ {A} ( \lambda ) = ( A- \lambda I ) ^ {-} 1$, called the resolvent of $A$( $\rho ( A)$ is called the resolvent set). With the help of resolvents a functional calculus for $A$ is built on functions analytic in a neighbourhood of $\sigma ( A)$:

$$f( A) = \frac{1}{2 \pi i } \int\limits _ \Gamma f( \lambda ) R _ {A} ( \lambda ) d \lambda ,$$

where $\Gamma$ is a contour enclosing $\sigma ( A)$( the unboundedness of $A$ imposes restrictions on the choice of $\Gamma$). Further conditions on the geometry of the spectrum and on the asymptotics of the resolvent enables one to extend this calculus.

The spectra of operator functions are defined by the formula

$$\sigma ( f( A)) = \{ {f( \lambda ) } : {\lambda \in \sigma ( A) } \}$$

(the spectral mapping theorem). The spectrum $\sigma ( A ^ {*} )$ of the adjoint operator coincides with $\sigma ( A)$ when $A$ is bounded; in general, $\sigma ( A ^ {*} ) \subset \sigma ( A)$.

If $\mathop{\rm dim} X < \infty$, then $\sigma ( A)= \sigma _ {p} ( A)$, and $X$ decomposes into the direct sum of subspaces invariant under $A$, on each of which $A$ induces an operator with one-point spectrum. Spectral theory of operators is concerned with finding infinite-dimensional analogues for this decomposition. See also Spectral analysis; Spectral synthesis; Spectral operator; Spectral decomposition of a linear operator.

References

 [1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) [2] T. Kato, "Perturbation theory for linear operators" , Springer (1980)