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Let $H$ be a complex, infinite-dimensional, separable Hilbert space. Let $T \in B(H)$ be an operator with disconnected spectrum. In the introduction of the paper: Jiang, C.; Sun, S.; Wang, Z. (1997). …

Let $H$ be a complex infinite dimensional separable Hilbert space. There are various extensions of the following well known result: Theorem (Lomonosov): Every nonscalar $T \in B(H)$ which commutes wi …

This question involves a possible method to prove the invariant subspace problem for (separable) infinite dimensional Hilbert spaces. The idea comes from various results on this topic; more precisely, …

Looking again at Solution 12.4 in Kubrusly's book, I have noticed that the proof can be used even to prove the statement above, with some small changes. I will briefly sketch such small modifications: …

Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book: Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert Space. CRC press above the statement of Th …

Let $H$ be a complex, infinite dimensional, separable Hilbert space. Fix any two nonzero operators $A,B \in B(H)$ such that $B$ is not a scalar multiple of $A$. It is well known that: $$ \| R_A (z) \| …

Let $X$ be a complex infinite dimensional Banach space, and let $T \in B(X)$ be nonscalar. The Fredholm spectrum of $T$ is defined by: $$ \sigma_{\Phi} (T) := \lbrace \lambda \in \mathbb{C} : T- \lamb …

In many cases, some problems are either solved in an affirmative way, or in a negative way. However, in some cases it turns out that some logical axioms lead to a proof of a certain statement, while o …