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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Properties of non-integer powers of the Hodge Laplacian

Consider a complete smooth Riemannian manifold $(M,g)$. I think that it is not difficult to prove that the $k$ Hodge Laplacian is essentially selfadjoint in the relevant $L^2$ space of $k$ forms, ...
V. Moretti's user avatar
2 votes
1 answer
298 views

Bounded operator on $L^2(\Bbb R^2)$

Let $\lambda\in \{z\in\Bbb C\mid \text{Re}(z)>0\quad \text{and}\quad \text{Im}(z)>0 \}$. Consider the operator $T_\lambda: L^2(\Bbb C)\to L^2(\Bbb C)$: $$ f\mapsto T_\lambda(f)(z)=\int_{\Bbb C}f(...
zoran  Vicovic's user avatar
8 votes
0 answers
405 views

Semigroups of matrices closed under conjugate transposition

An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
E W H Lee's user avatar
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Spectral measure for a finite set of mutually commuting normal operators

The following question is from Exercise $\S 11.11$ in A Course in Operator Theory written by John B. Conway: Suppose $\{N_1, \cdots, N_p\}$ is a finite set of mutually commuting normal operators in $...
Sanae Kochiya's user avatar
0 votes
0 answers
58 views

Explicit description for dual to operator space of Hilbert space

Let $H$ be a separable Hilbert space, and $B := \mathcal B(H)$ be the space of bounded operators on $H$. It is known that $B^\ast_\mathrm{strong} = B^\ast_\mathrm{weak}$ (see [Dunford, Schwartz, VI.1....
Vasily Ionin's user avatar
3 votes
0 answers
48 views

Norm under Gelfand map vs norm under left regular representation on $\ell^p$

Let $G$ be a discrete commutative group. Let $p \in [1,\infty)$ and consider the left regular representation $\lambda : \ell^1(G) \to \mathcal{B}(\ell^p(G))$; that is $\lambda(x)y := x*y$, where $$ (x*...
Leo Sera's user avatar
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2 votes
0 answers
58 views

Why is the essential numerical range defined as $W_e(T) = \bigcap_{K\in \mathcal K(H)} \overline{W(T+K)}$?

I have been introduced to the following definition of the essential numerical range of a bounded, linear operator on a separable, infinite-dimensional Hilbert space: $$W_e(T) := \bigcap_{K\in \mathcal ...
stoic-santiago's user avatar
2 votes
0 answers
55 views

Existence of a suitable smooth kernel

Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
Ali's user avatar
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1 vote
1 answer
118 views

Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$?

Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product $$X \otimes_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \...
J. De Ro's user avatar
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1 answer
119 views

Is $N(A^n)=N^n(A)\;$ for a self-adjoint operator $A$?

A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e....
Student's user avatar
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4 votes
2 answers
254 views

A space isometric to $\ell_\infty^2$

Consider a norm on $\mathbb C^2$ as $\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$ Question. Is $(\mathbb C^2,\|.\|)$ linearly isometric to $(\mathbb C^2,\|.\|_{\infty})$ where $\|(...
A beginner mathmatician's user avatar
2 votes
2 answers
266 views

When a quasinilpotent is nilpotent?

In the case of an infinite-dimensional complex banach space $X$, under what conditions can a quasinilpotent operator $T\in B(X)$ be determined to be nilpotent?
Phd m's user avatar
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1 answer
215 views

Showing a 2-by-2 matrix is a contraction

Let $S\subseteq\mathbb{T}:=\{z\in\mathbb{C}:\vert z\vert=1\}$ be a compact set such that $\operatorname{conv}S\supseteq\{z\in\mathbb{C}:\vert z\vert\leq\frac{1}{\sqrt{2}}\}$ and $B\in M_2(\mathbb{C})$....
Piku's user avatar
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1 vote
0 answers
21 views

Regularity of solutions of a 2nd order singular integro-differential operator

I have trouble finding the regularity of the solutions to a particular equation. I define $$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}...
BlueCharlie's user avatar
2 votes
1 answer
173 views

If $X$ is a Markov process, can we find a mild assumption ensuring that $\frac1t\operatorname E_x\left[\int_0^tc(X_s)\:{\rm d}s\right]\to c(x)$?

Let $(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$ for all $x\in E$ $\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$ $(\kappa_t)_{t\ge0}$ ...
0xbadf00d's user avatar
  • 161
3 votes
1 answer
121 views

Young-type inequality for bounded operator

Let $A$ and $B$ be two (non commuting) self-adjoint bounded operator acting on a Hilbert space and let $p,q>1$ such that $\frac1p+\frac1q=1$ Do we have a Young-type inequality such as $ \frac12|AB+...
Chris's user avatar
  • 301
3 votes
1 answer
424 views

Extension of a bounded linear functional

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B(\mathcal{H})}$ be an operator system. Suppose $T_n$ is a collection of all $n$-by-$n$ matrices equipped with ...
Piku's user avatar
  • 231
2 votes
1 answer
161 views

Extension of the projective norm to a cross norm

Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be an operator system. Is it possible to extend the projective norm (the greatest cross norm) ...
Piku's user avatar
  • 231
1 vote
0 answers
30 views

Limiting absorption principle for higher powers of resolvents

Let $H$, $A$ be self-adjoint operators on a Hilbert space. Moreover, let $I$ be a bounded open interval contained in the spectrum of $H$. Assume that $H$,$A$ satisfy the following positive commutator ...
Janik's user avatar
  • 141
1 vote
0 answers
248 views

Operators for norm for some classes of integral operators

Let $T:L^2(\mathbb{R}^n,\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n,\mathbb{R}^n)$ be a linear operator of the form: for all $x\in \mathbb{R}^n$ $$ T_{\kappa}(f)(x):=\int_{u\in\mathbb{R}^n} \, f(u)\...
ABIM's user avatar
  • 5,019
2 votes
0 answers
78 views

An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$

Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define $$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$ ...
ABB's user avatar
  • 3,898
1 vote
1 answer
157 views

Does Hörmander's condition imply smooth density of transition probabilities conditioned on non-blow-up?

Motivation. I’m not an expert on stochastic calculus and stochastic differential equations; I often see the Fokker-Planck equations and Hörmander's theorem formulated as addressing “transition ...
Julian Newman's user avatar
0 votes
1 answer
98 views

"Project" an operator outside of a von Neumann Algebra into it

Suppose $W$ is a proper von Neumann Algebra contained in $B(H)$ and the identity in $W$ is the identity mapping of $H$ (namely, $W$ does not have non-trivial null space). Given a self-adjoint $T\in W$...
Sanae Kochiya's user avatar
0 votes
0 answers
189 views

On the spectrum of a compact pertubation of a skew-adjoint operator

Let $A\colon \text{dom}(A)\subset H \to H$ ($H$ is a Hilbert space) be a skew-adjoint (i.e. $A^{*}=-A$), closed and densely defined operator. Then the essential spectrum is the set of spectral values $...
user99432's user avatar
  • 173
0 votes
1 answer
120 views

Diagonaloperators on $l^2(N)$ [closed]

Let $D$ be an operator defined by $D(c_n)_{n\in\mathbb{N}} = (a_n c_n)_{n\in\mathbb{N}}$. It's a well known fact that $D$ is well defined as an operator from $l^2(\mathbb{N})$ to $l^2(\mathbb{N})$ if $...
Matthias's user avatar
4 votes
1 answer
86 views

The adjoint of the Cesaro operator on $H^p$

For each $f=\sum_{k=0}^\infty b_kz^k\in H^p(\mathbb{D})$, $1<p<\infty$, we consider the following operators; $$ C(f)(z)=\sum_{n=0}^\infty \left(\frac{1}{n+1}\sum_{k=0}^n b_k\right)z^n $$ and $$ ...
Fernando Leon Saavedra's user avatar
2 votes
0 answers
192 views

Fourier transform of unbounded linear operator

I am trying to construct Fourier transform of a family of unbounded linear operators. Here is the construction. Fix $H$ a Hilbert space. Let $D\subset H$ be a fixed dense subset. Denote by $L(H)$ some ...
Ken.Wong's user avatar
  • 493
1 vote
0 answers
132 views

Description of state space of $C(K,M_n)$?

Edit: closed convex hull added. I am trying to understand the state space of $C(K,M_n)=C(K)\otimes M_n$ for $K$ a compact space. My guess would be that these are the closed convex hull of states on $C(...
C-star-W-star's user avatar
7 votes
0 answers
167 views

Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements

Consider the free product of $\mathbb{Z}/2$ with itself with generators $$ \mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle $$ and regard its group $C^*$-algebra $$ C^*(\mathbb{Z}/2*\mathbb{...
C-star-W-star's user avatar
1 vote
0 answers
88 views

2-positivity to 3-positivity

Let $B\in M_3(\mathbb{C})$ and $S_3= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix} $. Suppose $I_3\otimes I_2+B\otimes X+B^*\otimes X^*\geq 0$ for all $2$...
Piku's user avatar
  • 231
0 votes
1 answer
208 views

Semi-commutative von Neumann algebras

Suppose $\Omega$ is a $\sigma$-finite measure space with measure $\mu.$ Let $\mathcal M\subseteq B(H)$ be a von Neumann algebra. Can an element of $L_\infty(\Omega)\overline{\otimes}\mathcal M$ be ...
A beginner mathmatician's user avatar
4 votes
1 answer
241 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 ...
Andromeda's user avatar
  • 189
3 votes
1 answer
137 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 ...
Sijie Luo's user avatar
2 votes
0 answers
40 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 ...
alby's user avatar
  • 91
2 votes
1 answer
136 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 $(D,A)$ ...
ABB's user avatar
  • 3,898
0 votes
1 answer
52 views

Kernels of sequences of operators

Let $\left( S_{n}^{1}\right) $ and $\left( S_{n}^{2}\right) $ two sequences of operators in $\mathcal{L}(E_{1},F_{1})$ and $\mathcal{L}(E_{2},F_{2})$ where $E_{i},F_{i},i=1,2$ are Hilbert spaces such ...
Gustave's user avatar
  • 545
1 vote
0 answers
138 views

Generalized functional for solution of PDEs

Asked this on Math Stack Exchange awhile ago but it got ignored then deleted. To solve a differential equation of one variable, you need constraints equal to the number of derivatives. For a partial ...
Connor Dolan's user avatar
1 vote
0 answers
160 views

Discretizing a differential operator which is a function of the derivative operator

Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$. As an example, let $p(x)=x^{2}+2x$. Then ...
Mirar's user avatar
  • 350
4 votes
0 answers
245 views

Infinitesimal generator of a Markov process acting on a measure

Short version: The transition operator of a Markov process can act on measures (on the left) or functions (on the right). The infinitesimal generator acts on functions. Is there a way to understand ...
Ziv's user avatar
  • 121
2 votes
0 answers
166 views

Projections in von Neumann algebra tensor product

Let $\mathcal M\subseteq B(\mathcal H)$ be a von Neumann algebra with normal faithful semifinite trace $\tau.$ Consider the von Neumann algebra $\mathcal N:=L_\infty([0,1])\overline{\otimes}\mathcal M$...
A beginner mathmatician's user avatar
3 votes
1 answer
379 views

A trace inequality between self-adjoint operators

Let $A$ and $B$ be self-adjoint operators on some Hilbert space and $B$ is postive. Suppose we have $-B\leq A\leq B$.Is it true then that $\|A\|_p\leq\|B\|_p$ where $\|.\|_p$ is the Schatten-$p$ norm ...
A beginner mathmatician's user avatar
3 votes
0 answers
54 views

Injectivity of a convex combination of squares $A^*A$ in $\ell^\infty$

Consider two operators $A,B: \ell^p \to \ell^p$ ([defined and]bounded for all $p \in [1,\infty]$) as well as their adjoints $A^*,B^*:\ell^p \to \ell^p$. Assume $A^*A$ and $B^*B$ have trivial kernel ...
ARG's user avatar
  • 4,342
2 votes
1 answer
169 views

Non-injectivity of the limit of non-injective sequence of operators

It is known that the limit of a sequence of non injective operators is not necessarily non-injective, for instance, the operator \begin{eqnarray*} T_n &:&\ell ^{2}\rightarrow \ell ^{2} \\ x &...
Gustave's user avatar
  • 545
1 vote
0 answers
44 views

Examples of TRO $V $ and $C^{\ast} $-algebra $B $ for which $V\otimes^hB $ is a TRO

Let $V $ be a ternary ring of operator(TRO) and $B $ be a $\mathbb {C}^{\ast} $-algebra. Let $V \otimes^hB $ denotes the Haagerup tensor product of $V $ and $B $. Obviously if $V $ or $B $ is $\mathbb ...
Math Lover's user avatar
  • 1,065
1 vote
1 answer
115 views

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 ...
user332905's user avatar
3 votes
0 answers
76 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{...
moji's user avatar
  • 41
1 vote
0 answers
136 views

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?
user479310's user avatar
3 votes
2 answers
127 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 ...
Markus's user avatar
  • 1,361
1 vote
0 answers
113 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 $M(A)$)...
Andromeda's user avatar
  • 189
2 votes
1 answer
424 views

Structure of the inverse of a Fredholm integral operator of the second kind

NOTE: Cross-posted on Mathematics Stack Exchange I am trying to solve an equation of the form $$ (\mathbb{I} + K)\phi = f $$ where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\...
a1228d's user avatar
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