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

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Spectrum on an unbounded operator

Consider the operator $T_{c}=-\frac{d}{dx^{2}}+ c x^{2}$ with $c\in C^{*}$, $Re(c)>0$ defined on its domain $D_{c}=\{u\in L^{2}; T_{c}(u)\in L^{2}\}$. Put $c=a+ib$ avec $a>0$ et $b\in R$. ...
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39 views

A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n

In my research of operator algebras and their connection with machine learning I of course use the well know result: For the map $ tr:M_n \to M_n $ denoting the transpose map of matrices (meaning ...
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101 views

The C*-envelope of the algebra of continuous functions on a compact topological space is commutative

In my research in operator theory, specifically in C* algebras and enveloping, I came across this strange footnote in a text (locally published in non English where I study) which states the following:...
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I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
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Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators

Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...
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Equivalence between complex and real operator norms

Consider a real $m\times n$ matrix $A$ and the $p$-norms in $\mathbb{C}^n$ and $\mathbb{C}^m: \|x\|=\left(\sum|x_i|^p\right)^{1/p}$. One defines the real $p$-norm of $A$ as $\|A\|=\sup\frac{\|Ax\|}{\|...
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quasi-nilpotent part of a dual operator

Definitions and notation. Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as \begin{equation*}H_0(T):=\left\{...
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153 views

Convergence of functionals on compact projections on a separable Hilbert space

Let $H$ be a separable Hilbert space over $\mathbb{C}$, say $\ell_2$ for simplicity. Let $\mathcal{K}(H)$ denote the space of all compact operators on $H$ and $\mathcal{P}(H)$ the set of all finite ...
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Generalized singular numbers and the Haagerup $L^p$ spaces

Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$. The $L^p$ norm on $M$ is given by \begin{...
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89 views

almost invariant half space for a dual of a restricted operator

Let $X$ be an infinite-dimensional Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. A closed subspace $Y$ of $X$ is said to be an almost-invariant halfspace (...
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Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs. I'm reading Stochastic Differential Equations in ...
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1answer
64 views

$C(X)$-compact operators and families of compact operators

In this question Operators on Hilbert $C^*$-module and families of Fredholm operators I asked about the relation between being a family of compact operators $F:X \to K(H)$ on Hilbert space $H=\ell^2$ ...
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149 views

Convergence in trace

Let $A$ and $B$ be two self-adjoint, positive definite Compact operators on a Hilbert space $\mathcal{H}$. Further, let $A$ be trace class. Define $C_n \equiv AB(\frac{I}{n} + BAB)^{-1}$. Does $\frac{...
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55 views

Operators on Hilbert $C^*$-module and families of Fredholm operators

If $A$ is a $C^*$-algebra, there is a notion of Hilbert $A$-module (which is something like Hilbert space but the inner product takes values in $A$). The standard example is $H_A:=\{(a_n)_{n=1}^{\...
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Multivariable Weighted shift and subnormality

I have asked this question in mathstackexchange but didn't get any answer. I hope, I will get my answer here. Let $\mathbb B^m$ denote the Euclidean ball in $\mathbb C^m.$ Does there exist a ...
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103 views

Convergence of unitary products on a Hilbert space [migrated]

First: I'm sorry for the basic question--I can move it to Math SE if necessary... Let $X$ be a Hilbert space and suppose $\{U_k\}_k$ is a sequence of unitary operators on $X$. Let $\|\cdot\|$ be the ...
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When is a linear operator on $C^{0,\alpha}(\overline{\Omega})$ a multiplication?

The title says it all, really. Suppose that I have a linear operator $T$ from $C^{0,\alpha}(\bar{\Omega})$ into itself, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^d$ (e.g. the unit ball ...
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2answers
91 views

Behavior of orbits under small perturbations

Perhaps this question is too easy for mathoverflow, at least this is how it seems, but I got no answer on stackexchange. Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a ...
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166 views

analytic continuation argument

In "Pseudo-spectra, the harmonic oscillator and complex resonances" (login required), the author says Sections $2$ and $3$ of this paper concern the operator $Hf(x)=(-\frac{d^{2}}{dx^{2}}+cx^{2})...
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Abstract Wave Equation and Semigroups

If an operator $A$ on a Hilbert space $H$ generates a strongly continuous semigroup, does then the operator $B$ on $H \oplus H$ given by the matrix $$ B := \begin{pmatrix} 0 & \mathrm{id} \\ A &...
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123 views

The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant

In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on: If $ \mathbb{H} $ is an RKHS and we denote the ...
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extension for a complex operator

Let be $\lambda>0$. Put $$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial \overline{z}}-z\frac{\...
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positive operator surjectivity

In nonlinear analysis and monotone operator theory it is well known that if the operator $A$ is a maximal monotone and strongly monotone on a real Hilbert space $H$, then $A$ is surjective. This can ...
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68 views

Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates an analyitc ...
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198 views

When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space. Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
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375 views

What is the group of automorphisms of $l^{\infty}$?

What is the group of automorphisms of $l^{\infty}$? I think it would be the permutations of the integers. Is this right?
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260 views

Compact non-nuclear operators

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, non-...
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135 views

The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras

Is there a name for the following property of a $C^{*}$ algebra $A$? $$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$ Example of this situation is $A=C(X)$ where $X$ is the ...
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121 views

Does a $W^*$ envelope exist?

I know that an operator algebra $A$ has a "minimal" $C^*$-algebra $C$ containing $A$, which is known as the $C^*$ envelope of $A$. The existence of such a minimal $C^*$-algebra generated by $A$ (a ...
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A problem from Sakai's book on derivations on C(K) and differential structure on K

In his book, Operator Algebras in Dynamical Systems, at page 59 Sakai poses the following question. Problem: Let K be a compact space and suppose that C(K) has a non-zero closed *-derivation. Then ...
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136 views

Infinitesimal generator and stationarity

The following question is bothering me. I think it is probably known but I cannot find any reference... Let $(X_t)$, $(Y_t)$, $(Z_t)$ denote 3 Feller processes with respective infinitesimal ...
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152 views

Unconditionally $p$-converging operators on $L_{1}[0,1]$

Let $1\leq p<\infty$. We say that an operator $T:X\rightarrow Y$ is unconditionally $p$-converging if $T$ takes a weakly $p$-summable sequence to a norm null sequence. Question: Is every ...
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57 views

What equals $\ker[(A-\lambda I)^+]$ for a negative unbounded operator $A$?

We have the following result: $\{ E_{\lambda}; \, -\infty <\lambda < + \infty\}$ is a spectral family, where $E_{\lambda}$ is the projection of $H$ onto the null space $\mathscr N \left(A_\...
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1answer
43 views

Simplify the expression of $ T^+$ for an unbounded operator $T$?

For a negative unbounded operator $T$, what equals the operator $$ T^+ = \left[\frac{1}{2}(|T| + T) \right]^{**},$$ where $|T|= (T^2)^{1/2}$ and $A^{**} $ is the minimal closed extension of an ...
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Are Ritt operators mean ergodic?

In the following, $T$ is a bounded operator on a Banach space $X$. $T$ is called "power bounded" if $\sup_{n\in \mathbb N}\|T^n\|<\infty$; $T$ is called "mean ergodic" if the Cesàro sums $\frac{1}...
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Spectra of Dirac operators

1. Suppose that $M$ is a spin manifold. The spin structure of $M$ is not uniquely defined: in other words, $M$ may have many nonequivalent spin structures. For each choice of spin structure there is ...
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Specific type operators and basic sequences

Let $s$ be the space of rapidly decreasing sequences, i.e. $s=\{\xi=(\xi_j)_j\colon\,\,\sup_j|\xi_j|j^n<\infty\,\,\text{for all}\,\,n\in\mathbb{N}\}$ and $s'$ its topological dual, i.e. $s'=\{\eta=(...
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Strongly continuous semigroups and symbols of pseudo differential operators

I am considering the Cauchy IVP for the evolution equation $$u_t + \Psi u =0$$ where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$. The ...
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Chord-arc property of n-tuples of commuting operators

Assume that we have an $n$-tuple $S^0=(S^0_1,\dots,S^0_n)$ of commuting operators in a Hilbert space $H$ and another such an $n$-tuple $S^1$. Is it possible to connect these two $n$-tuples by a ...
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192 views

On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces

I am presently doing research concerned with operator algebras and operator theory and I thought to write here in the hopes of seeking expert advice on an idea I had here. The classic Sz.-Nagy ...
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Density of sets whose image is dense

This is probably easy, but I can't think of an answer. Assume $X$ is a Banach space and $A$ is a (not assumed closed) subspace of $X$. Let $T:X \to X$ be a bounded linear operator, which is also ...
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Help in understanding result from publication on operator theory

in my research on dilations of contractions on Hilbert spaces and manifolds I have come across this nice publication concerning the classic Sz-Nagy theorem on the Arxiv by Levy and Shalit which states ...
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549 views

Are Hilbert-Schmidt operators on separable Hilbert spaces “Hilbert Schmidt” on the space of Hilbert Schmidt Operators?

Let's consider a separable Hilbert space $(\mathcal H, \langle\cdot, \cdot\rangle_{\mathcal H})$ with Norm $||\cdot||_{\mathcal H} := \langle\cdot, \cdot\rangle^{1/2}_{\mathcal H},$ orthonomal basis $(...
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Is this a characterization of commutative $C^{*}$ algebras?

Assume that $A$ is a $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A_{sa} \iff ba \in A_{sa}$$ Is $A$ necessarily a commutative ...
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Spectrum of compact operator between different Banach spaces

Let $X, Y$ be two different Banach spaces, and let $T: X \to Y$ be a compact linear operator. Suppose the identity $I : X \to Y$ is well-defined. (For example, we could have $X = L^2([0,1])$ and $Y = ...
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213 views

Diagonalizable unitary operators [closed]

Let $u\colon H\to H$ be a unitary operator on a separable Hilbert space $H$ and let $(e_n)_n$ be a fixed orthonormal basis in $H$. Is it possible to decompose $u$ as $u=v^*dv$ where $v$ is a unitary ...
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Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...
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132 views

Continuity of the largest eigenvalue with respect to length

Let $k:\mathbb{R}^+\to\mathbb R^+$ be a continuous function. For $a>0$, define $T_a$ acting on $L^2[0,a]$, by $$T_af(x) = \int_0^a k(|x-y|)f(y)\,dy.$$ Clearly for each $a>0$, the operator $T_a$...
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Eigenspace of a specific operator

Consider the operator $T:\ell^\infty({\mathbb N})\to\ell^\infty({\mathbb N})$ defined by $$ (Tx)_m=\sum_{k=m+1}^\infty p_{k,m} \ \ x_k, $$ where $$ p_{k,m}=\frac k{(k-1)(k-m)(k-m+1)}. $$ Then $T$ is a ...
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Uniform continuity of spectrum as function of operator [closed]

It is well known that the spectrum is continuous as function of operator. More precisely, let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of linear ...