[Paper Review] On the sum between a closable operator $T$ and a $T$-bounded operator

Dan Popovici, Zoltán Sebestyén+1 more|arXiv (Cornell University)|Sep 19, 2014

Spectral Theory in Mathematical Physics9 references3 citations

TL;DR

This paper develops a perturbation theory for closable unbounded operators on Hilbert spaces by introducing a novel framework based on the range properties of 2×2 matrix operators of the form $\begin{pmatrix} I & A \\ -B & I \end{pmatrix}$. The key contribution is a generalized perturbation theorem showing that if $T$ is closable and $S$ is $T$-bounded with specific range and duality conditions, then $T + S$ is closable and its adjoint equals $T^* + R$ for a suitable $R$, extending classical results of Hess–Kato, Kato–Rellich, and Wüst under weaker assumptions.

ABSTRACT

We provide several perturbation theorems regarding closable operators on a real or complex Hilbert space. In particular we extend some classical results due to Hess--Kato, Kato--Rellich and Wüst. Our approach involves ranges of matrix operators of the form $\left(\begin{array}{cc}\!\! I & A\!\!\\ \!\! -B& I\!\!\end{array} ight)$.

Motivation & Objective

• To generalize classical perturbation theorems for unbounded operators by weakening the standard $T$-boundedness conditions.
• To establish sufficient conditions under which the sum of a closable operator $T$ and a $T$-bounded operator $S$ remains closable.
• To characterize the adjoint of $T + S$ via the range of matrix operators involving $I$, $T$, $S$, and their adjoints.
• To unify and extend results of Hess–Kato, Kato–Rellich, and Wüst in a common framework using operator matrix ranges.
• To provide conditions ensuring that $T + S$ is essentially selfadjoint or selfadjoint under weaker assumptions than previously known.

Proposed method

• Utilizes the range of the matrix operator $\begin{pmatrix} I & A \\ -B & I \end{pmatrix}$ as a central analytical tool to characterize domain and adjoint properties.
• Applies a duality lemma (Lemma 2.1) to deduce dense definition and adjoint relations from range and duality conditions.
• Establishes that if the range of $\begin{pmatrix} I & A \\ -B & I \end{pmatrix}$ is dense or closed, and duality holds, then $A$ and $B$ are adjoints.
• Introduces a key technical condition: $\|Sx\|^2 \leq \|Tx\|^2 + \|x\|^2$ for $x \in \operatorname{dom}T$, which generalizes $T$-boundedness.
• Uses the matrix operator framework to derive conditions under which $T + S$ is closable and $ (T + S)^* = T^* + R $ for a specific $R$.
• Applies the theory to symmetric, selfadjoint, and essentially selfadjoint operators, deriving generalized versions of classical perturbation theorems.

Experimental results

Research questions

• RQ1 Under what conditions is the sum $T + S$ of a closable operator $T$ and a $T$-bounded operator $S$ itself closable?
• RQ2 How can the adjoint of $T + S$ be characterized when $T$ is closable and $S$ is $T$-bounded?
• RQ3 Can the classical Hess–Kato, Kato–Rellich, and Wüst perturbation theorems be extended to less restrictive assumptions using matrix operator ranges?
• RQ4 What role does the range of the matrix operator $\begin{pmatrix} I & A \\ -B & I \end{pmatrix}$ play in ensuring the closability and adjoint structure of $T + S$?
• RQ5 Under what conditions is $T + S$ essentially selfadjoint or selfadjoint when $T$ is essentially selfadjoint or selfadjoint?

Key findings

• If $T$ is closable and $S$ is $T$-bounded with $\|Sx\|^2 \leq \|Tx\|^2 + \|x\|^2$ for all $x \in \operatorname{dom}T$, and the matrix operator $\begin{pmatrix} I & T^* \\ -T & I \end{pmatrix}$ has dense range, then $T + S$ is closable and $ (T + S)^* = T^* + R $ for a specific $R$.
• The perturbation theorem extends the Hess–Kato result: if $\|Sx\|^2 \leq \|Tx\|^2 + \|x\|^2$ and $\|S^*k\|^2 \leq q\|T^*k\|^2 + \|k\|^2$ with $q < 1$, then $T + S$ is closable and $ (T + S)^* = T^* + S^* $.
• For essentially selfadjoint $S$ with $\operatorname{dom}S \subseteq \operatorname{dom}T$ and $\|Th\|^2 \leq \|Sh\|^2 + \|h\|^2$, if $T$ is symmetric on $\operatorname{dom}S$, then $S + T$ is essentially selfadjoint.
• If $S$ is selfadjoint and $\|Th\|^2 \leq q\|Sh\|^2 + \|h\|^2$ for $q < 1$ and $T$ symmetric on $\operatorname{dom}S$, then $S + T$ is selfadjoint.
• The closedness of $B^* + R$ is guaranteed if $R$ is $B^*$-bounded with $B^*$-bound less than 1, which ensures the validity of the perturbation result.
• The range condition $\operatorname{ran}\begin{pmatrix} I & A \\ -B & I \end{pmatrix} = \mathscr{H}_4 \times \mathscr{H}_3$ is sufficient to deduce that $D^* = A$ and $C^* = B$ under duality and domain compatibility.