To show an element in the sprectum of a matrix $M$ is negative in the spectrum of $-M$

Question

I am trying to solve the following problem:

Let $V$ be a Banach space , $\lambda \in \mathbb{C}$, an operator $H \in \mathbb{B}(V \oplus V)$ in a form of \begin{align} H = \begin{pmatrix} 0 & x \\ y & 0 \end{pmatrix} \end{align}

Show that $\lambda \in \Lambda(H)$ if and only if $-\lambda \in \Lambda(-H)$.

(Hint: To find an operator $T$ such that $THT^{-1} = -H$)

$\lambda \in \Lambda(H)$ => $-\lambda \in \Lambda(-H)$

If $\lambda \in \Lambda(H)$ then $H - \lambda I$ is not invertible, and $(H - \lambda I)T = 0$ for some nonzero $T$. That is, $H T = \lambda T$, $- H T = - \lambda T$. I have totally no idea to continue. Can anyone help me?

Answer 1

This is elementary: an operator $T$ is invertible if and only if $-T$ is invertible. Now

$H-\lambda I$ is not invertible if and only if $(-H)-(-\lambda)I$ is not invertible

Therefore $\lambda$ belongs to the spectrum of $H$ if and only if $-\lambda$ belongs to the spectrum of $-H$.

For a proof with the operator $T$ such that $THT^{-1}=-H$, I suggest $$ T=\begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix} $$ where $I$ is the identity on $V$.