Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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771 views

Compact sets as point spectrum of a bounded operator

It is well known that if $K$ is any compact set in $\mathbb{C}$, then there exist a bounded linear operator $T:l_2\to l_2$ such that $\sigma(T)=K$. My questions are: Q1) Does there exist $T$, a ...
0
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1answer
816 views

Operator Norm and Hilbert-Schmidt

I am trying to prove $\|T\| \leq \|T\|_{HS}$ I understand everything up until the following two lines, could somebody please explain why $\| Tx \| \leq \|T \|_{HS}$ $\|x\|$ implies that $\| T ...
3
votes
1answer
196 views

Strongly-Continuous linear functionals on $\mathcal{B}(H)$

Suppose $H$ is a complex Hilbert space and $$w: \mathcal{B}(H) \longrightarrow \mathbb{C}$$ is a bounded linear functional on $\mathcal{B}(H)$ such that $w$ is continuous even if $\mathcal{B}(H)$ is ...
2
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1answer
183 views

Operator Norm $\| T\|$

Would somebody mind explaining why if $T$ is a continuous and bounded operator on a Hilbert space $H$, we have $$\|T\| = 1 \;\;\;\Rightarrow \;\;\;\|Te_n \| = \|e_n\|\;\;\mbox{for all }\;\;x\in H$$ ...
1
vote
1answer
311 views

Compact operator? self adjoint operator? Stirling's formula

Define a pair of operators $S\colon\ell^2 \rightarrow \ell^2$ and $M\colon\ell^2 \to\ell^2$ as follows: $$S(x_1,x_2,x_3,x_4,\ldots)=(0,x_1,x_2,x_3,\ldots) $$ and ...
0
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1answer
150 views

Help to find all of the eigenvalues of the operator $T:L^2([0,1]) \rightarrow L^2([0,1])$

Define $K: [0,1] \times [0,1]\rightarrow \mathbb{R}$ by $$K(x,y) =\begin{cases} (1-x)y &\text{if } 0 \le t \le x\\ (1-y)x& \text{if }x \le y \le 1\end{cases}$$ Also, Consider the ...
4
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1answer
154 views

$T(V)$ is a closed subspace of $V$?

Let $V$ be a normed vector space (not necessarily a Banach space) and let $S$ and $T$ be continuous linear transformations from $V$ to $V$. If we assume that $T=T \circ S \circ T$. Then how to show ...
2
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1answer
125 views

Linear image of closed convex subset

Let $X$ and $Y$ be two Banach spaces and $A:X\rightarrow Y$ a linear, continuous map. Let $M\subseteq X$ be a closed, convex subset of the unit sphere in $X$. When is $A(M)\subseteq Y$ closed?
3
votes
2answers
323 views

Examples of not completely bounded maps

Let $\phi:\mathcal{A}\longrightarrow\mathcal{B}$ be a bounded map between $C^*$ algebras. $\phi$ is said to be completely bounded if the natural extension map \begin{eqnarray} ...
2
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1answer
338 views

Properties of self-adjoint operator

I am struggeling with an easy deduction that I cannot see for some reason today myself, hope to get some help on this: Suppose $T$ is a compact self-adjoint operator on a Hilbert space $\mathcal{H}$. ...
7
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1answer
703 views

Is the right shift operator bounded?

I was reading my lecture notes for functional analysis when I came across the following statement: Let $(e_{n})$ be a total orthonormal sequence in a separable Hilbert space H. The right shift ...
1
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1answer
350 views

The exponent of self-adjoint operator

If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is it necessarily that $A^2$ is self-adjoint as well?(admittedly, $A^2$ is densely defined)
7
votes
1answer
423 views

Is this operator compact?

Suppose ($x_n$) is a normalized, linearly independent, sequence in a reflexive Banach space $X$, and $T$ is an injective, strictly singular, bounded operator on $X$ such that $Tx_n\longrightarrow ...
4
votes
1answer
685 views

Compact multiplication operators

In class, we started talking about operators on Banach spaces after covering the Arzela-Ascoli Theorem. We defined a continuous operator $T\colon X \to Y$ to be compact if $\overline{T(B_X)}^{Y}$ is ...
0
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1answer
364 views

Relation between range and kernel of a linear operator

Let $S = I - T$ where $T$ is a compact linear operator on a Hilbert space $H$. Why is it that the range of $S$ is equal to $S((\ker S)^{\perp})$?
0
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1answer
416 views

Are there non-commuting self - adjoint operators?

I am currently working through a set of lecture notes on operator theory. For self - adjoint operators, I just showed that, if $B_1$ and $B_2 \in \mathcal{B}(\mathcal{H},)$ are self - adjoint then ...
0
votes
1answer
244 views

compact operator and its spectrum

I'm trying to study compact operators, but i'm having a little trouble with the 'practice'.. What are some tecniques to prove an operator compact. I know it can be shown that a limit of finite range ...
2
votes
1answer
83 views

meaning of this operator ??

given the operator $ P_{\Lambda } = (f\in L^{2}(R)^{even}| f(q)=0 , |q| \ge \Lambda) $ what does it mean? the operator $ P_{\Lambda} $ acts over a function $f(q) $ by setting this (even) function to ...
3
votes
2answers
242 views

eigenfunctions of the adjoint of an operator

If the eigenfunctions of a linear operator are known, is there a way to calculate the eigenfunctions of the corresponding adjoint operator based on the known eigenfunctions? In other words, what's the ...
2
votes
0answers
162 views

Find the minimum value of the maximum eigenvalue of operator A?

So we are given the following: Operator $A$ with $Au=-u''$; $u \in D_A = \{u\colon[a,b]\rightarrow R,u\in C^2([a,b]),u(a)=u(b)=0\}$; $D_A$ is dense in $L^2((a,b))$. Find the minimum value that is ...
2
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1answer
160 views

Proving $aa^*\geq0$ in a $C^*$ algebra

I tried to construct a proof of above statement by using matrices. It goes like this :- Let $a$ be any arbitrary element in a $C^*$ algebra. Consider the matrix $\begin{pmatrix}0 & a\\a^* & 0 ...
2
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1answer
184 views

Adjoint identity

I want to show that $\operatorname{Range}(A^*)^\perp \subset \operatorname{Null}(A)$ where $A:E \supset D(A) \to F$ is an unbounded closed linear operator densely defined in $E$, and $E$ and $F$ are ...
2
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1answer
1k views

Taylor expansion for matrices

Is it possible to define a Taylor expansion for matrices ? Can I use functional derivative ? More precisely I have to calculate something like : $\ln(A+B)$ using a Taylor expansion, where $A$ and $B$ ...
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1answer
117 views

What kind of objects act on bounded linear operators?

I am currently studying for a first course in operator theory, and I was wondering about the following that occurred to me whilst doing my reading: It seems to me that, by talking about maps in ...
2
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1answer
119 views

Characterizations of the form domain for unbounded selfadjoint operators

This question follows from this one and especially from Willie Wong's answer: link. In Reed & Simon's book Methods of modern mathematical physics, vol. I, pag.277, the form domain of a ...
4
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0answers
125 views

Relations between spectrum and quadratic forms in the unbounded case

Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
8
votes
4answers
336 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty ...
6
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1answer
349 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
4
votes
2answers
81 views

if $A^k$ is compact for some positive $k$, is it possible that $A$ is not bounded?

If $A$ is a linear operator on a normed vector space $X$ and $A^k$ is compact for some positive integer $k$, is it possible that $A$ is unbounded? If not, how to prove that $A$ is bounded?
2
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0answers
234 views

Cauchy's integral formula for operators

I study this article : A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model. Massimo Campanino and Abel Klein. Comm. Math. Phys. 104 ...
3
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2answers
102 views

Operators with complemented range

Holub proved that Fredholm operators are stable under compact perturbations. I am interested in a slight refinement of this theorem. Suppose we have two operators $T_1$ and $T_2$ acting on a primary ...
2
votes
2answers
761 views

Coercivity vs boundedness of operator

The definition of coercivity and boundedness of a linear operator $L$ between two $B$ spaces looks similar: $\lVert Lx\lVert\geq M_1\lVert x\rVert$ and $\lVert Lx\rVert\leq M_2\lVert x\rVert$ for some ...
1
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1answer
155 views

Boundedness of supremum of an Integral operator

I am trying to find an $L_2$ - bound on a certain class of operators, and on my way I produced an estimate for which I need to show that \begin{equation} \sup_{x \in \mathbb{R}^n} \, ...
1
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0answers
85 views

positively homogeneous asymptotic expansion associated to the symbol of a pseudodifferential operator

I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic ...
1
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1answer
134 views

Sum of bounded and unbounded operators

Is there a Banach space $X$, $S$ an unbounded operator defined on a dense subspace $D$ of $X$ and a bounded operator $T$ on $X$ such that $$S+T|_D$$ is bounded? What if $T$ is assumed to be ...
1
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1answer
43 views

What is a semibounded polynomial on $\mathbb{R}^n$?

I am stuck with the following expression, because no google search gives an answer to my problem. Here it is: I am reading a text that states "Let a(x) be a semibounded from below polynomial on ...
0
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1answer
121 views

Strict coisometries and operator norm.

I got stuck at the following problem. Let $X,Y$ be normed spaces. A bounded linear operator $\tau\in\mathcal{B}(X,Y)$ is called strictly coisometric if $$ ...
7
votes
3answers
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Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?

Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain $$f(x+a) = ...
13
votes
1answer
1k views

How to prove that an operator is compact?

Consider $T\colon\ell^2\to\ell^2$ an operator such that $Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
3
votes
1answer
601 views

Under which circumstances is the Laplacian compact?

I want to know when the Laplacian is a compact Operator. Do you know some good literature about this topic? For instance, is the Laplacian compact on the Sobolev space $H^2(\Omega)$? Or maybe on the ...
2
votes
1answer
503 views

Commutativity of Convolution in higher dimensions

I have a basic question about how to show that convolution in dimension $n$ is commutative - or maybe it is rather a question about change of variables .. So on $\mathbb{R}$ I know how to show ...
3
votes
2answers
104 views

Understanding an example of a Distribution

Whilst reading the article about restrictions of distributions (generalized functions) on Wikipedia (here) I had trouble understanding the example of a distribution defined on the subset $V = (0,2) ...
6
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2answers
249 views

Proving $A: l_2 \to l_2$ is a bounded operator

Let us consider the following linear operator acting on $l_2$: $$ A(x_1,x_2,x_3,\ldots) ~\colon=~ \left(x_1,\frac{x_1+x_2}{2},\frac{x_1+x_2+x_3}{3},\ldots\right) $$ I need to show that $A$ is a ...
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0answers
93 views

Introductory book on Distribution theory [duplicate]

Possible Duplicate: Distribution theory book Is there a good alternative to Friedlander Introduction to the Theory of Distributions ? Many thanks !
3
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2answers
129 views

Binet's Formula. An operational approach.

I read quite a while ago this proof of Binet's formula. ( I am not 100% sure this is the way it was presented, but it gives an idea. I'm not approving of this method or saying it is correct.) Let ...
2
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0answers
168 views

What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
5
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1answer
866 views

Linear transformations in infinite dimensional vector spaces

If we look at an $n$ - dimensional vector space $V$ and a linear transformation \begin{equation} T : V \to V, \quad x \mapsto Tx \quad \forall \, x \in V \end{equation} then given a choice of basis ...
3
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1answer
132 views

Help understanding a proof on Taylor's formula in Schwartz space $S(\mathbb{R}^n)$

I am having trouble understanding a proof to establish a specific version of Taylor's formula. I'll first give the statement and then below cite the part where I am stuck, so here is what I'd like to ...
3
votes
2answers
273 views

Asymptotic Expansion for heat operator $e^{-t\triangle}$

I'm afraid the question below might turn out to be very stupid - I just don't know how to make sense of two asymptotic expansions, given the heat operator $e^{-t\triangle}$ with $\triangle$ a ...
3
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1answer
369 views

Eigenvalues of compact operators and his adjoint.

Let $T: H \to H$ be a compact operator with $H$ a Hilbert space. Let then $\lambda \neq 0$ be an eigenvalue of $T$ with eigenfunction $v$. Is then $\lambda$ an eigenvalue for the adjoint $T^*$ ...