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Let A be a quasinilpotent operator on a Hilbert space and let every operator of the algebra generated by $A$ and $A^{*}$ have finite spectrum. Does then follow, that A is nilpotent ? See also quasi …

The answer is yes : Let $h$ be an even real valued Schwartz function whose Fourier transform has compact support. Then choose $f(y) = \int_{-\infty}^y x h(x)^{2} dx$ .

I found a counterexample : Let $e_{1},e_{2},...$ be ON basis of the Hilbert space and define A by $Ae_{2n-1} = \sqrt{1-\frac{1}{n^{2}}} \ e_{2n} \ + \ \frac{1}{n} \ e_{2n+1}$ , $\ \ $n=1,2,3,... …

Let A be a quasinilpotent operator on a Hilbert space and let $A^{*}A$ have finite spectrum. Does then follow, that A is nilpotent ?

The second resolvent equation you are studying is an important tool in quantum scattering theory. See M. Reed and B. Simon, "Methods of Modern Mathematical Physics", Vol IV, Academic Press

How can one calculate the index of a Fredholm operator numerically ? In numerically calculations one uses always finte dimensional spaces. But linear operators on finite dimensional spaces have alway …

Let $H$ be a Hilbert space and $U$ a closed subspace of $H\times H$ . Does then exist closed subspaces $V$ and $W$ of $H$ such that $H\times H = U \oplus (V\times W)$ ? See also Perturbations of an …

In http://www.jstor.org/pss/2047905 you can find a weighted shift operator that has this property.

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator). Is this also true in the Solovay model …

Counterexample for $n = 2$ : Let $A_k$ be the orthonormal projection on the span of $$(\cos(2 \pi (k-1) / 5), \sin(2 \pi (k-1) / 5))^\mathsf{T} , \quad k = 1...5.$$ Then $k(A_k,A_l) = \vert \cos(2 \pi …

For Hilbert spaces, the conjecture follows from fact 4 and the answer to question Complement of a subspace which is a cartesian product applied to the kernel of the map $H\times H\ni (v,w) \mapsto Av …

The answer is no : Let $U_1 = I$, where $I$ is the identity and $U_2$ linear independent to $I$ such that $U_2^* + U_2 \ge 0$ . Then choose $\phi(U_1) = I$ and $\phi(U_2) = -I$ .

Here a simple example : Let X be the cartesian product of $L^{\infty}$ and $L^{1}$ on the interval $[0,1]$, let $P_{t}$ the canonical projection on the subspace of functions with support $[0,t]$ and …

No, choose $\Omega=\{z \in \mathbb{C} : |z| \leq 1 \} ,\ f(z)=Re\ z^{n}$ and let $n\rightarrow \infty$ .