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

learn more… | top users | synonyms

1
vote
0answers
109 views

On isolated points of the approximate point spectrum of a bounded operator

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$. Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively. Let ...
1
vote
0answers
70 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 ...
0
votes
0answers
52 views

inverse of operator

I want to calculate the inverse of the operator $T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial ...
1
vote
1answer
97 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 ...
3
votes
1answer
168 views

Self-adjointness of the components of the magnetic derivative

On $L^{2}(\mathbb{R}^{n})$ define the operator $\Pi_{j} u := (-i\partial/\partial x_{j} - A_{j})u$, where $A_{j} \in L^{2}_{loc}(\mathbb{R}^{n})$ represents the $j$-th component of the magnetic ...
0
votes
1answer
104 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), ...
2
votes
0answers
120 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 ...
0
votes
0answers
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 ...
3
votes
1answer
58 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 ...
0
votes
0answers
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 ...
0
votes
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 ...
2
votes
0answers
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 ...
5
votes
0answers
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 ...
2
votes
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 ...
2
votes
0answers
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. ...
4
votes
0answers
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
votes
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 ...
1
vote
2answers
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 ...
5
votes
2answers
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 ...
3
votes
2answers
526 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 ...
5
votes
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 ...
52
votes
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 ...
0
votes
1answer
90 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 = ...
1
vote
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 ...
5
votes
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 ...
23
votes
0answers
472 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 ...
0
votes
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 ...
4
votes
2answers
269 views

The closure of all periodic homeomorphisms of circle

Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...
3
votes
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 ...
0
votes
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 ...
1
vote
0answers
124 views

Generating a series representation for the inverse of the operator $f(f)$

I am considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
8
votes
1answer
216 views

Hodge de Rham operator and orientability

Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...
5
votes
0answers
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 $S$ ...
1
vote
0answers
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. $f.g\in X$. ...
1
vote
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 ...
0
votes
0answers
37 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 ...
6
votes
1answer
499 views

An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$ where these integers are ...
3
votes
2answers
191 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 ...
3
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
10
votes
1answer
556 views

Is there an algebra for divergent series summation operators?

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, which is either one ...
4
votes
1answer
228 views

Fredholm subvector spaces of $B(\mathcal{H})$

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e_i\}\;\;i\in \mathbb{N}$. Definition: We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if ...
4
votes
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 $z=x+ i y \in \mathbb C$ ...
4
votes
1answer
106 views

Restriction of a semigroup to a form domain

Say, we have a Hilbert space $H$ with a semibounded self-adjoint operator $A:D(A)\to H$ generating a strongly continuous semigroup $T(t):H\to H$. Is it possible to restrict $T(t)$ to a form domain of ...
4
votes
0answers
120 views

Noncommutative Poincaré inequalities

This is a question on how (or if) people in the community think about the Poincaré inequality in noncommutative geometry. In geometry, the Poincaré inequality (when it exists) gives a bound on a ...
14
votes
0answers
440 views

Unital $C^{*}$ algebras which all elements have path connected spectrum

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. Is the tensor product of two path connected algebra, a path ...
2
votes
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 ...
11
votes
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 ...
2
votes
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 $A^nx$ ($n\in\mathbb N$) ...