0
votes
2answers
227 views

Spectral decomposition of compact operators

Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an ...
0
votes
2answers
83 views

Does positivity preserve compactness? [closed]

Suppose $A$ and $B$ are operators on a (separable) Hilbert space $H$ and $A \leq B$. Is it true that if $B$ is compact then $A$ is compact too? If not, could you please show a counterexample?
1
vote
2answers
192 views

Characterisation of compact operators

It is known (and easy) to prove that if $T: H\longrightarrow H $ is compact, where $H$ is a Hilbert space, then for any orthonormal basis $ e_n $ we have $||Te_n||\longrightarrow 0$. My question is ...
6
votes
1answer
184 views

Is there a nice “minimum” of two symmetric operators?

Let $A$ and $B$ be two bounded symmetric positive operators in Hilbert space, such that $A-B$ is trace class. If needed, $A$ and $B$ may be assumed reasonably "small", let's say, Hilbert-Schmidt. ...
1
vote
2answers
196 views

Finite dimensional approximations of operators on Hilbert spaces

Let $e_1,e_2,\dots$ be a Schauder basis for a Hilbert space $(V , \langle \cdot , \cdot \rangle)$. Let $A:V \to V$ be an operator. Finally, let $V_n = {\rm span}( e_1, \dots, e_n)$. Let $i_n : V_n ...
1
vote
1answer
121 views

Special kind of operators

Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation $$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$ where ...
3
votes
1answer
156 views

On the self-adjoint part of a quasinilpotent operator

Disclaimer: this is not research-level, but I've read some non research-level questions/answers on quasinilpotent operators here, some of them involving renowned users. So I thought I'd give it a try. ...
0
votes
2answers
582 views

When are two operators simultaneously diagonalisable?

I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...
6
votes
1answer
219 views

A doubt about the parts of the spectrum of tensor products

Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, ...
0
votes
1answer
670 views

Can we construct a Hilbert space where the operator following differencial operator is symmetric?

I'd like to know if one can define a pertinent Hilbert space where the operator $$A_p v := -\frac{1}{2} v" + (vF + v\int_\mathbb{R} Sp + p\int_\mathbb{R} Sv )'$$ is symmetric. Here, $p$ satisfies ...
4
votes
3answers
424 views

Inequality of von Neumann for more than two contractions

Good morning, I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized ...