On the sum between a closable operator $T$ and a $T$-bounded operator

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 $\left(\begin{array}{cc}\!\! I & A\!\!\\ \!\! -B& I\!\!\end{array}\right)$.

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