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 \c …

An bouneded linear operator $A \in L(X, Y)$ ($X$, $Y$ Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that $$ \left( \sum_{j=1}^N \| A x_j\|_X^2 \righ …

"In this section, we deal with positive linear maps $\phi : A \rightarrow M$ between two unital C∗-algebra $A$ and $M$ with units denoted by $I$. In fact, we may assume that $A$ is …

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 though …

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, w …

I have already ask this question in here without any response. How to express $(S_{\alpha}(t)){t \geq 0}$ where $S{\alpha}(t)=e^{t^{\alpha} A}$ as a "one parameter semigroup" a …

I am trying to find out the essence of what a determinant is. Besides, in finite dimensions, determinant is the kind of numerical invariant that determines the invertibility of a l …

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 …

"This was until a beautiful ‘index-theoretic’ characterisation of quasitriangularity was obtained by Apostol, Foias and Voiculescu, which had the unexpected consequence that if an …

The operator $E$ is defined as $Eu_n=u_{n+1}$. I encountered a strange equality. when I tried out Let $u_n$ represent a series such that $$u_{n+2}=u_{n+1}+u_n. \tag{$\star$} …

Consider the Sobolev space $W^{k,p}(\Omega)$ for $k\in \mathbb N$, $p\in [1,\infty]$ and some open domain $\Omega\subset \mathbb R^n$ $^*$. Then it is known that $W^{k,p}(\Omega)$ …

What are the basic examples of functions in hardy space and bergman space that I can test the theorems against.I want to know how to produce examples for each parameter. Edited: I …

I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011). There, they work on a Hilbert s …

Let $X$ and $Y$ be Banach spaces and $T\in B(X,Y)$ be Fredholm. Then there is $S\in B(Y,X)$ such that $ST=I+K_{1}$ and $TS=I+K_{2}$ where $K_{1},K_{2}$ are compact operators.Proof …

Consider $C_0$-semigroup $S_t:\mathscr{B(H)} \to \mathscr{B(H)}$ with generator $U$. Now define $P_t:\mathscr{B_1(H)} \to \mathscr{B_1(H)}$ where $P_t(\rho)=S_t\rho S_t^*$. How ca …