While reading the book,"Hilbert Space Operators in Quantum Physics" by Jiří Blank, Pavel Exner, Miloslav Havlíček. I come across the statement.
"In an infinite–dimensional Banach space $X$ the two expressions are no longer equivalent: the inverse $(T −\lambda)^{−1}$ exists if $λ$ is not an eigenvalue of $T$, but it may be neither bounded nor defined on the whole $X$."
Could you explain the statement?
My attempt to understand the statement:- If $X$ is finite dimensional, Then I know that $X$ is a Banach space. The inverse $(T −\lambda)^{−1}$ exists if $λ$ is not an eigenvalue of $T.$ I am not able to identify the two statements correctly.$ I am not able to understand why they are not equivalent in infinite dimensional space. Could you expand bit?
Let $X=\ell^2(\mathbb{N})$ and $$T(x_1,x_2,\ldots, x_n,\ldots )\\ =(0,x_1,{1\over 2}x_2,\ldots,{1\over n}x_n,\ldots )$$ Then $T$ is injective. The inverse is given by $$T^{-1}(y_1,y_2,\ldots y_n,\ldots )\\ =(y_2,2y_3,\ldots,ny_{n+1},\ldots )$$ is defined on $$Y=\left \{y\in \ell^2(\mathbb{N})\,:\, y_1=0,\ \sum_{n=2}^\infty n^2|y_n|^2<\infty \right \}$$ Thus the inverse is defined on a proper, even non dense, subspace and the inverse is unbounded because $T^{-1}e_{n+1}=ne_n,$ where $e_n$ denotes the sequence with entry $1$ at the $n$th position and $0$ at other positions. Summarizing all features mentioned in OP are present for $\lambda=0.$