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

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93 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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102 views

adjoint of the operator rotation [closed]

I need to calculate the adjoint of the operator $T_{a}=a i(x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x})$ with $i^{2}=-1, \quad a\in \mathbb{C}$ and domain $D=\{\varphi\in L^{2}(R), ...
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27 views

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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118 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 ...
3
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1answer
57 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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55 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 ...
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1answer
34 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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83 views

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 ...
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80 views

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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30 views

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. ...
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1answer
74 views

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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38 views

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 ...
4
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2answers
182 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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214 views

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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172 views

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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525 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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1answer
168 views

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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89 views

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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1answer
208 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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471 views

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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1answer
126 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 ...
5
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1answer
162 views

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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1answer
108 views

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 ...
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95 views

Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows: Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...
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54 views

Can a semigroup be defined on a Banach algebra? [closed]

I simply need to know that how a semigroup of operators (say $\{T(t)\}_{t\geq 0}$) is defined on any Banach algebra (say $X$)? For $(f,g\in X)$ now the so called product is also there i.e. $f.g\in X$. ...
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1answer
113 views

When is a homogeneous polynomial an inner function on the unit torus?

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...
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36 views

Kernel for projection operators onto the spaces of piecewise linear loops

For every $m\in\mathbb{N}_+$, $k=0,\dots,m-1$, denote $I_{m,k}:=(\frac{k}{m},\frac{k+1}{m})$. Denote $W = H^1(S^1,\mathbb R^{2n})$ = The Hilbert space of $C^1$ maps maps $v(t)$ from the circle to ...
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2answers
190 views

Weak convergence implies norm convergence for trace class operators?

It is known that weak convergence implies norm convergence in $\ell^1(\mathbb{N})$, see e.g. here. Because of the typical analogies of the Schatten ideals $C_p \subset B(H)$ (where $H$ is a Hilbert ...
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46 views

Matrix representation of linearized PDE

My motivation for this question is to investigate the linear stability of steady solutions of a nonlinear PDE by computing eigenvalues of the linearized equations. Specifically, I have a function ...
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0answers
77 views

Heat semigroup ultracontractive?

Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
52
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2answers
810 views

Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...
4
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3answers
174 views

Show that the Laplacian operator on the Heisenberg group is negative

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
2
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2answers
266 views

Weakly compact operators between Banach spaces

Let $X$ and $Y$ be complex Banach spaces and $B(X,Y)$ be the Banach space of all bounded operators. An operator $T\in B(X,Y)$ is weakly compact if $T(\{ x\in X;\; \| x\| \leq 1\})$ is relatively ...
2
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1answer
88 views

Pointwise convergence of polynomials to a function on a compact set K that is 1 on some disc D and zero outside D

Motivation of my question: Let $A$ be a bounded selfadjoint operator with spectral measure $E$ and $x$ a vector. Then it is easily seen that the closed linear span of all $A^nx$ ($n\in\mathbb N$) ...
9
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1answer
101 views

A generalisation of $C_0$-semigroups

A $C_0$-semigroup is a strongly continuous family $\{T(t)\}$ of bounded linear operators on a normed space $X$, indexed in $\mathbb R_+$ and with two additional properties that make it look like an ...
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3answers
682 views

Bibliographic request concerning an article by Bernstein and Robinson

Concerning the article "Bernstein, Allen R.; Robinson, Abraham. Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in finding ...
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1answer
129 views

How do we know the map is $w^{*}$-continuous?

I am reading a paper by David Blecher, which contains the following: " If $T: Y \to Z$ is a surjective isometric module map between $W^{*}$-modules over $M$, then $T$ is unitary. Also, $T$ is a ...
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105 views

Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...
3
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2answers
659 views

Can one hear the shape of a drum for operators?

M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ...
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1answer
87 views

Approximation of the central support

Let $(M,\tau)$ be a tracial von Neumann algebra, i.e. a unital subalgebra $M=M''\subset \mathbb{B}(H)$; a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ ...
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1answer
139 views

Invariant subspaces are reducing subspaces in $L^2(\mu)$; where $\mu$ is a singular measure w.r.t Lebesgue measure

I have already posted this question on math.stackexchange but didn't get any answer. I hope this is the right place to ask this question. Recently I was reading a book "Operator Function and system" ...
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74 views

One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
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73 views

Holomorphic natural bundles and operators

I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting. After a quick thought, I've gone through the standard ...
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68 views

Spectral theory of Bochner integral operators

Consider the following (somewhat simplified) situation. Let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of bounded linear operators acting on ...
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1answer
88 views

The monotone operator in $BV$ space

I am considering the following minimizing problem: $$ \min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\} $$ where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...
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1answer
162 views

comparing norms of tensor product of two Hilbert spaces

Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$ consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...
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Factorization of the Fokker-Planck semigroup

I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...
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1answer
100 views

Comparison between spectra

Let $G$ be a normal operator with compact resolvent on a Hilbert space $H$ such that ${\rm ker}(G) \neq {0}$. Further let $P$ be the orthogonal projection onto ${\rm ker}(G)$, and let $G_{0}:=G+P$. ...
3
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1answer
125 views

$M_Λ(A) → A ⊗ M_Λ(C)$

I saw this in here. Let $A$ be a Banach algebra, and let $\Lambda$ be a non-empty set. We denote by $M_\Lambda(A)$ be the set of $\Lambda\times\Lambda$ matrices $(a_{ij})_{i,j\in\Lambda}$ with entries ...
3
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227 views

Graded structures for simple $C^{*}$ algebras without nontrivial idempotent

Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded ...