Let σa p(A) denote the approximate point spectrum for an...

Let $\sigma_{a p}(A)$ denote the approximate point spectrum for an operator $A \in \mathscr{B}(\mathscr{H})$. (a) Show that $\Pi_{j=1}^{n}\left(A-\lambda_{j} I\right)$ is bounded below on $\mathscr{H}$ if and only if $A-\lambda_{j} I$ is bounded below for $1 \leq j \leq n .$ (b) Show that for any polynomial $p, \sigma_{a p}(p(A))=p\left(\sigma_{a p}(A)\right)$. Does the analogous result, with "'approximate point spectrum" replaced by "point spectrum" hold? The point spectrum of $A$ is $\{\lambda: \operatorname{ker}(A-\lambda)$ is nontrivial $\}$, i.e., the set of eigenvalues of $A$.

Let $\sigma_{a p}(A)$ denote the approximate point spectrum for an operator $A \in \mathscr{B}(\mathscr{H})$.
(a) Show that $\Pi_{j=1}^{n}\left(A-\lambda_{j} I\right)$ is bounded below on $\mathscr{H}$ if and only if $A-\lambda_{j} I$ is bounded below for $1 \leq j \leq n .$
(b) Show that for any polynomial $p, \sigma_{a p}(p(A))=p\left(\sigma_{a p}(A)\right)$.
Does the analogous result, with "'approximate point spectrum" replaced by "point spectrum" hold? The point spectrum of $A$ is $\{\lambda: \operatorname{ker}(A-\lambda)$ is nontrivial $\}$, i.e., the set of eigenvalues of $A$.

Key Concepts

Bounded Below Operators

A bounded below operator on a Hilbert space is one for which there exists a constant c > 0 such that the norm of Ax is at least c times the norm of x for all vectors x. This concept is crucial because it implies that the operator is injective and has closed range, and it plays a key role in the analysis of operator invertibility and stability under perturbations.

Approximate Point Spectrum

The approximate point spectrum of an operator consists of those scalars for which the operator, when subtracted by that scalar times the identity, fails to be bounded below. Equivalently, these are the values for which there exist sequences of unit vectors that are almost annihilated by the operator. This concept extends the idea of eigenvalues to include limit points and is fundamental in understanding the spectrum in infinite-dimensional settings.

Polynomial Functional Calculus and the Spectral Mapping Theorem

Polynomial functional calculus is a tool that allows one to apply polynomials to operators in a well-defined manner. The spectral mapping theorem asserts that the spectrum of a polynomial in an operator is the polynomial applied to the spectrum of the operator. This principle is important in operator theory because it links algebraic transformations of operators to transformations of their spectra, and similar properties hold for approximative spectra under certain conditions.

Point Spectrum

The point spectrum, defined as the set of eigenvalues of an operator, consists of those scalars for which the operator fails to be injective. While every eigenvalue is in the approximate point spectrum, the reverse inclusion does not necessarily hold. This distinction is significant in spectral theory and in understanding the full structure of an operator’s spectrum on infinite-dimensional spaces.

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