Properties of Spectra

If $\operatorname {spec} A$ is the spectrum of $A$ (the set of complex numbers $\lambda$ such that $A - \lambda$ is not invertible), if $\Pi (A)$ is the set of complex numbers $\lambda$ such that $A - \lambda$ is not bounded from below, and if $\Gamma (A)$ is the set of complex numbers $\lambda$ such that the closure of the range of $A - \lambda$ is a proper subspace of $\mathbf{H}$ (i.e., distinct from $\mathbf{H}$), then $\operatorname {spec} A = \Pi (A) \cup \Gamma (A)$. The set $\Pi (A)$ is called the approximate point spectrum of $A$; a number $\lambda$ belongs to $\Pi (A)$ if and only if there exists a sequence $\left\{ f_n\right\}$ of unit vectors such that $\left\| (A-\lambda )f_n\right\| \to 0$. An important subset of the approximate point spectrum is the point spectrum $\Pi_0(A)$; a number $\lambda$ belongs to it if and only if there exists a unit vector $f$ such that $Af = \lambda f$ (i.e., $\Pi_0(A)$ is the set of all eigenvalues of $A$). The set $\Gamma (A)$ is called the compression spectrum of $A$.

An assertion such as that if $A$ is an operator and $p$ is a polynomial, then $\operatorname {spec} p(A) = p(\operatorname {spec} A)$ is called a spectral mapping theorem; other instances of it have to do with functions other than polynomials, such as inversion, conjugation, and wide classes of analytic functions.

$\Pi (A)$ is the set of complex numbers $\lambda$ such that $A - \lambda$ is not bounded from below (the approximate point spectrum); $\Pi_0(A) \subset \Pi (A)$ is the set of complex numbers $\lambda$ such that $Af = \lambda f$ for some unit vector $f$ (the point spectrum, i.e., the set of eigenvalues of $A$); $\Gamma (A)$ is the set of complex numbers $\lambda$ such that the closure of the range of $A - \lambda$ is a proper subspace of $\mathbf{H}$ (the compression spectrum). Together, $\operatorname {spec} A = \Pi (A) \cup \Gamma (A)$.

Two operators $A$ and $B$ are similar if there exists an invertible operator $P$ such that $P^{-1}AP = B$.

$\Pi (A)$ is the set of complex numbers $\lambda$ such that $A - \lambda$ is not bounded from below (the approximate point spectrum); $\Pi_0(A) \subset \Pi (A)$ is the set of complex numbers $\lambda$ such that $Af = \lambda f$ for some unit vector $f$ (the point spectrum, i.e., the set of eigenvalues of $A$); $\Gamma (A)$ is the set of complex numbers $\lambda$ such that the closure of the range of $A - \lambda$ is a proper subspace of $\mathbf{H}$ (the compression spectrum). Together, $\operatorname {spec} A = \Pi (A) \cup \Gamma (A)$.

If $A$ and $B$ are operators, and if at least one of them is invertible, then $AB$ and $BA$ are similar (for the proof, apply $BA = A^{-1}(AB)A$ in case $A$ is invertible, or $AB = B^{-1}(BA)B$ in case $B$ is). Since similar operators have the same spectrum (Problem 75), this implies that if at least one of $A$ and $B$ is invertible, then $AB$ and $BA$ have the same spectrum. In the finite-dimensional case more is known: with no invertibility assumptions, $AB$ and $BA$ always have the same characteristic polynomial. If neither $A$ nor $B$ is invertible, then, in the infinite-dimensional case, the two products need not have the same spectrum, but their spectra cannot differ by much.

$\Pi (A)$, the approximate point spectrum of $A$, is the set of complex numbers $\lambda$ such that $A - \lambda$ is not bounded from below; equivalently, $\lambda \in \Pi (A)$ if and only if there exists a sequence $\left\{ f_n\right\}$ of unit vectors such that $\left\| (A-\lambda )f_n\right\| \to 0$.

$\Pi (A)$, the approximate point spectrum of $A$, is the set of complex numbers $\lambda$ such that $A - \lambda$ is not bounded from below; equivalently, $\lambda \in \Pi (A)$ if and only if there exists a sequence $\left\{ f_n\right\}$ of unit vectors such that $\left\| (A-\lambda )f_n\right\| \to 0$.