On direct sums and operator spectra Let $H$ be a Hilbert space, and $\{H_{n}\}$ be a sequence of mutually orthogonal closed subspaces of $H$ such that $H = \oplus_{n=1}^{\infty} H_{n}$. For each $n$, we have a bounded operator $A_{n}$ on $H_{n}$  such that $\{A_{n}\}$ is uniformly bounded. It's given that there exists a positive real $c$ such that for all $n \in \mathbb{N}$, $\lambda \in \sigma(A_{n})$ $\implies$ $|\lambda| \geq c$, that is, the spectra of $A_{n}$ are uniformly bounded away from zero. Is the sequence $\{A_{n}^{-1}\}$ uniformly bounded? I could prove it for normal operators, because for them, the spectral radius coincides with the norm, but I am struggling to prove it without that assumption. Tried to find a counterexample too, but didn't get anything concrete. Any help would be immensely appreciated.
Regards,
Saptak
 A: Consider the Hilbert space $H=\ell^2(\mathbb{N})$ and denote the canonical orthonormal basis by $\{\delta_n\}_{n=1}^{\infty}$. Consider the sequence of subspaces defined by
\begin{equation} 
H_1=\{\delta_1\},\quad H_2=\{\delta_2,\delta_3\},\quad H_3=\{\delta_4,\delta_5,\delta_6\},\quad H_4=\cdots.
\end{equation}
Let $A_n$ be the operator on $H_n$ whose matrix representation with respect to the canonical orthonormal basis of $H_n$ is given by
\begin{equation}
 J_n:=I_n+N_n=\begin{pmatrix}  1 & 1 & & & \\  & 1 & 1 \\ & & ... & ... \\ & & & 1 & 1 \\ & & & & 1  \end{pmatrix} \in \mathbb{C^{n \times n}},
\end{equation}
where $I_n$ denotes the $n\times n$ unit matrix.
Then $\sigma_{H_n}(A_n)=\{1\}$ for all $n\in \mathbb{N}$ and
\begin{align}
\|A_n\|^2&=\sup_{\|x\|_2=1,x\in \mathbb{C}^n} \|J_nx\|^2 \\
&\leq \sup_{\|x\|_2=1} |x_1+x_2|^2+\cdots +|x_{n-1}+x_n|^2+|x_n|^2\\
&\leq \sup_{\|x\|_2=1} 4(|x_1|^2+|x_2|^2)+\cdots+4(|x_{n-1}|^2+|x_{n}|^2)+|x_n|^2\\
&\leq 8.
\end{align}
On the other hand, it follows from this post that
$$J_n^{-1}=I_n-N_n+N_n^2-\cdots +(-1)^{-n+1}N_n^{n-1}.$$
Since the powers of $N_n$ can be easily computed, we see that the last column of this matrix consists only of ones with alternating signs, which implies that
$$\|J_n^{-1}\|^2\geq n .$$
