# Questions tagged [operator-theory]

Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

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### Norm inequalities for sums of two basic elementary operators

I'm working on this paper. What I'm interested in is this theorem: where $M_{A,B}(X)=AXB$ I don't know why we can find those two sequences $X_n$ and $x_n$, either I'm finding difficulties to show the ...
198 views

### Takesaki lemma 1.16 (volume II, chapter VII)

I am trying to understand the proof of the implication $(i)\implies (ii)$ in Takesaki's book "Theory of operator algebras II", chapter VII, which says the following: The relevant setting ...
64 views

### Embedding finite dimensional subspaces of Schatten p-classes

I have recently read a paper by Talagrand: Embedding subspaces of $L_{1}$ into $l^{N}_{1}$, and part of the chapter: Finite dimensional subspaces of $L_{p}$, written by Jonson and Schechtman. The ...
30 views

### Potential scattering for non-decaying potential

I am reading this note on potential scattering by Siu-Hung Tang and Maciej Zworski. Just wondering if there is a scattering theory for the Schrödinger with a non-decaying potential $- \Delta + V$ on ...
93 views

### On the eigen vectors of a diagonalizable matrix

Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$. Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
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1 vote
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### Regarding variation of spectra

I have been reading the article The variation of spectra by J.D Newburgh. in this article and all related reference/ articles, the term 'variation of spectra' keeps coming in, but I nowhere find a ...
68 views

### Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$

My research needs help in finding examples of unitary matrices $U$ which have entries \begin{align} U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
1 vote
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### Extreme points of the unit ball of bounded operators on $L^p(\mathbb{R}_+)$

Is it known what the set of all extreme points of $B(L^p(\mathbb{R}_+))$ (bounded operators on $L^p(\mathbb{R}_+)$) is?
97 views

### Unicellular compact operators

An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...
1 vote
108 views

### Is it true that $\omega = \sum_{(k,l)\in I^2}\omega(p_k - p_l)$

Let $\{p_i\}_{i \in I}$ be a collection of projections in a $C^*$-algebra $A$ such that $\sum_{i \in I} p_i = 1$ in the strict topology (note here that $1$ is the unit of the multiplier algebra $M(A)$)...
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### Relaxation of notion of positive definite function

A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
158 views

### Takesaki: Lemma about enveloping von Neumann algebra

Consider the following lemma with proof from Takesaki's book "Theory of operator algebra I" (p121): It appears to me that Takesaki claims at the end of the proof that $\pi(A)_1$ is $\sigma$-...
130 views

### Takesaki's proof of the Kaplansky density theorem

Consider the following fragment from Takesaki's book "Theory of operator algebra I": Why is the boxed sentence true? It looks like they replace $A$ by its strong$^*$-closure. Is this ...
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### Stinespring's theorem: can we choose the dilation to be an isometry?

Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
1 vote