(An Inversion Spectral Mapping Theorem.) Suppose that A is...

(An Inversion Spectral Mapping Theorem.) Suppose that $A$ is an invertible operator in $\mathscr{B}(\mathscr{H})$. The goal of this problem is to show that $$ \sigma\left(A^{-1}\right)=\left\{\frac{1}{\lambda}: \lambda \in \sigma(A)\right\} $$ (a) Show that if $A-\lambda I$ is not bounded below, then $A^{-1}-\frac{1}{\lambda} I$ is not bounded below, and conversely that if $A^{-1}-\mu I$ is not bounded below, then $A-\frac{1}{\mu} I$ is not bounded below. Show that the eigenvalues of $A^{-1}$ are precisely the reciprocals of the eigenvalues of $A$. (b) Show that $A-\lambda I$ fails to have dense range in $\mathscr{H}$ if and only if $A^{-1}-\frac{1}{\lambda} I$ fails to have dense range in $\mathscr{H}$. Exercise $5.8$ may be helpful here. Conclude that $$ \sigma\left(A^{-1}\right)=\left\{\frac{1}{\lambda}: \lambda \in \sigma(A)\right\} $$ This result holds more generally for any invertible element in a unital Banach algebra; see for example, p. 204 in [8].

(An Inversion Spectral Mapping Theorem.) Suppose that $A$ is an invertible operator in $\mathscr{B}(\mathscr{H})$. The goal of this problem is to show that
$$
\sigma\left(A^{-1}\right)=\left\{\frac{1}{\lambda}: \lambda \in \sigma(A)\right\}
$$
(a) Show that if $A-\lambda I$ is not bounded below, then $A^{-1}-\frac{1}{\lambda} I$ is not bounded below, and conversely that if $A^{-1}-\mu I$ is not bounded below, then $A-\frac{1}{\mu} I$ is not bounded below. Show that the eigenvalues of $A^{-1}$ are precisely the reciprocals of the eigenvalues of $A$.
(b) Show that $A-\lambda I$ fails to have dense range in $\mathscr{H}$ if and only if $A^{-1}-\frac{1}{\lambda} I$ fails to have dense range in $\mathscr{H}$. Exercise $5.8$ may be helpful here.
Conclude that
$$
\sigma\left(A^{-1}\right)=\left\{\frac{1}{\lambda}: \lambda \in \sigma(A)\right\}
$$
This result holds more generally for any invertible element in a unital Banach algebra; see for example, p. 204 in [8].

Key Concepts

Spectrum

In operator theory, the spectrum of an operator consists of all scalars for which the operator - multiplied by the identity - fails to be invertible. It captures values where the operator behaves atypically, influencing the resolvent and the operational calculus, and it is central to understanding the operator's properties and behavior.

Invertible Operator

An invertible operator is one that has a bounded inverse, meaning there exists another operator that undoes its action. In spectral theory, the invertibility of an operator is closely related to its spectrum excluding zero, and it allows for the examination of relationships between the operator and its inverse through mappings such as inversion.

Spectral Mapping Theorem

The spectral mapping theorem links the spectrum of a function applied to an operator to the image of the operator's spectrum under that function. For example, when taking the reciprocal of an invertible operator, this theorem indicates that the spectrum of the inverse is the set of the reciprocals of the original spectrum, highlighting a profound relationship between functional calculus and spectral properties.

Bounded Below Operator

An operator is bounded below if there exists a positive constant such that the norm of the operator acting on any vector is at least this constant times the norm of the vector. This concept is significant when analyzing the stability of the operator's behavior and is used to ascertain when an operator and its inverse maintain or fail certain boundedness properties related to their spectral elements.

Dense Range

An operator has a dense range if the closure of its range equals the whole space. This property is crucial in spectral analysis because it implies that the operator nearly covers the entire space, and its failure can indicate eigenvalues or other spectral issues. The dense range property is employed to compare the behavior of an operator and its inverse, particularly in regards to their spectral characteristics and surjectivity.

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