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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Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
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1answer
30 views

Approximations of compact operators

Let $(\xi_n)_{n=1}^\infty$ be a sequence in a Hilbert space $K$ convergent to some $\xi$. Suppose we have a compact operator $T$ on $K$ such that $T\xi = 0$. Can we find a sequence of compact ...
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36 views

Canonical Forms For Matrices

In the following paper by Wedderburn what are the restrictions on the field $\mathbb F$ or on the linear application $\varphi$ that the author refers to obtain the matrix B? ...
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31 views

Number Operator closable on Fock Space?

In Bratelli Robinson the number operator in Fock space is defined as: $$\mathcal{D}(N):=\{\phi\in\mathcal{F}:\sum_{n=1}^\infty n|\|\phi_n\|<\infty\}\\ N:\mathcal{D}(N)\to ...
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1answer
50 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
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12 views

Differential Formula Simplification

Define operators $x,D,1$ by $xf=xf$, $Df=\frac{d}{dx}f=f'$, and $1f=f$. Notice, then that $$(x+D)^nf=\sum_{k=0}^np_k(x)D^kf,\ \ \ \ \ \ \ f\in R[x],$$ for some sequence of polynomials ...
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2answers
27 views

Bounded, surjective linear operator between Banach spaces

How can I show that for a given surjective linear operator $T: X \to Y$ between Banach spaces, if there exists an $\epsilon > 0$ such that $||Tx|| \geq \epsilon||x||$ for all $x \in X$, then $T$ is ...
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28 views

A puzzling derivation about the expectation of [$\hat{X}$, $\hat{H}$]

a free particle of mass $m$, with Hamiltonian $\hat{H} = \frac {\hat{P}^2} {2m}$, where $\hat{P} = -i \hbar \frac{\partial} {\partial x}$. The commutative relation is given by $[\hat{X}, \hat{H}] ...
4
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1answer
52 views

Convergence of operator

I would like to know how to solve the following problem (since I didn't manage to solve it on today's exam): Let $A_h:L^1(a,b)\to L^1(a,b)$ be defined: $$A_h f(x)=\frac{1}{h}\int_x^{x+h} g(t) dt,$$ ...
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18 views

Question about the norm of the bounded inverse of a closed operator

Suppose $A$ is a closed operator (not necessarily bounded) in a Banach space $X$ with bounded inverse $A^{-1}$. Suppose $\mu>\frac{1}{\|A^{-1}\|}$. The question is to show the existence of a vector ...
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2answers
46 views

Spectrum of a bounded operator $T$ satisfying $T^n=I$

Let $\mathcal{H}$ be an infinite dimensional Hilbert space, suppose $T\in \mathcal{B}(\mathcal{H})$ is a bounded operator and suppose that $n$ is the smallest natural number so that $T^n=I$. Let ...
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40 views

Find the adjoint operator.

Consider the sequence space $\ell_p$ and S defined by $(1\leq p<\infty)$$$ S:\ell_p\to\ell_p:(x_1,x_2,x_3,\ldots)\mapsto(0,x_1,x_2,\ldots) $$ Find the $S^*$ operator.
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2answers
43 views

Computing the norm of operator when space is equipped with sup norm and $L^1$ norm

Let $\phi $ be the linear functional $\phi (f)=f(0)-\int^1_{-1}f(t)\:\mathrm{d}t$ a.Compute the norm of $\phi$ as a functional on Banach space $C[-1,1]$ with sup norm. b.Compute the of $\phi$ as a ...
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3answers
83 views

What is a predual of the Banach space of compact operators on $\ell^2$?

I am wondering if the space $K(\ell^2)$ of compact operators on $\ell^2$ can have a predual. Thank you in advance for your help.
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39 views

definition of unitary operator

Wiki says " A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $U^{*}U=UU^{*}=I$ , where $U^{*}$ is the adjoint of $U$, $I$ is the identity ...
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31 views

Using Dirac Delta with functions and derivative operators

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 ...
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1answer
33 views

Spectral radius of an operator equals its norm

Let $X$ be a Banach space and $A:X\to X$ a bounded operator. We know that the spectrum of $A$ is always included in the ball $B(0,|A|)$ and the spectral radious $r(A)$ is the smalest radius such that ...
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39 views

Spectrum of a bounded operator and Liouville's theorem

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We know that the spectrum of $A$ is not empty, because otherwise we find a contradiction by using the holomorphy of the function ...
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0answers
41 views

What's the name of this type operator?

If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$. If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ ...
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1answer
43 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
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1answer
40 views

Riesz Lemma for reflexive spaces

I know the proof of Riesz Lemma: Let $Y$ be a closed (proper) subspace of a normed space $X$. Let $\varepsilon >0$. Then it exists an element $x \in X$ such that $||x||=1$ and $d(x, Y) \geq ...
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1answer
14 views

Give an example of a spanning set of $\ell^2(N)$ which is also a Bessel sequence but not a frame for $\ell^2(N)$

We know that in a finite dimensional Hilbert space, every spanning set is a frame, but this is not true for infinite dimensional space. It is easy to find an example which is a spanning set but not a ...
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1answer
19 views

Extension of a linear operator

Let $T$ be a linear operator defined on the space of the algebraic polinomials in $[0,1]$ (polinomials with rational coefficients) such that for each $k \in \mathbb{N}, T[x^k]=0$. Is it possibile to ...
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21 views

Existence of a lim

For a net of bounded operators $A_\alpha$, when we can say its limit exists? i.e. when does $\lim A_\alpha $ exist? For example I know that the limit of a net of projections exists. but I think in ...
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26 views

What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
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25 views

Proving something is a convolution operator…

If we define the operator $K(a)=F^{−1}aF$ where $ F:L^2({\mathbb R})\to L^2({\mathbb R})$, is the fourier transform given by $$\left(Ff\right)\left(x\right)=\int_{{\mathbb ...
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22 views

Question about convergence of sum

Let $T\in B(H,E)$ where $H$ a seperable hilbertspace, $E$ a seperable Banach space. By parsevals identity $$\left\|T^*\right\|^2= \sup_{ \left\|x^*\right\|\leq 1}\left\|T^*x^*\right\|^2 = \sup_{ ...
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23 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
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44 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ ...
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1answer
53 views

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

Let $X$ be a complex Hilbert space, and let $A$ be a bounded linear operator on $X$. Define the real part of $A$ to be $\Re(A)=\frac{1}{2}(A^{\star}+A)$, and define ...
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1answer
37 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
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2answers
21 views

How to show something is a convolution operator?

I have the operator $W(a)$ defined by $$W(a)=F^{-1}aF$$ where $F$ denotes the fourier transform and $a$ is a function on $L^{\infty}$. I need to prove that this is convolution operator, but I don't ...
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2answers
23 views

What are non-tagential limits?

I'm reading this article where they use a set of functions, $H^{\infty}$, defined like this "Let $H^{\infty }$ be the closed subalgebra of $L^{\infty }({\mathbb R})$ that consists of all functions ...
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1answer
50 views

Relation between $A^{*}B=B^{*}A$ and $AB^{*}=BA^{*}$

Let $A$ and $B$ be two matrices. Can we say $A^{*}B=B^{*}A$ implies $AB^{*}=BA^{*}$? how about when $A$ or $B$ are normal? Any comments could be useful. Thanks.
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1answer
41 views

Sufficient condition for two operators being identical on Hilbert space

Considering two bounded linear operators $S,T$ in $\mathcal{B}(X)$, where $X$ is a complex Hilbert space. If $\def\norm#1#2{\langle {#1},{#2}\rangle} \norm{Sx}{x} = \norm{Tx}{x}$ for all $x\in X$, do ...
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1answer
16 views

Reference for projective limits of Banach Spaces..

I have just started studying a book about pseudo-differential operators and I came across projective limits of Banach spaces and I got lost. I have never studied this, so can anyone recommend me some ...
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45 views

Find norm of linear operators

I have to check if those operators are bounded and if so what are their norms. 1) $\phi:C^1[0,1]\ni f > \rightarrow\int_0^{1/2}f(t)dt+f'(2/3)\in\mathbb{R}$ with norm ...
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1answer
81 views

Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1 $ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
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1answer
39 views

Exponential map commutes with the projection homomorphism into Calkin algebra

Let $B(X)$ denote the space of bounded operators on a Banach space $X$ and $K(X)$ the compact operators. Let $\pi : B(X) \to B(X) / K(X)$ denote the quotient map. Let $u$ denote the right shift ...
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1answer
25 views

About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
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1answer
15 views

Fredholm index of invertible bounded operator

Let $X$ be a Banach space and $T: X \to X$ be bounded and invertible. Is it true that the Fredholm index $\mathrm{ind}(T) = 0$?
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1answer
63 views

“right shift” il $L^1$

Let $X=L^1(\mathbb{R})$ be the space of Lebesgue integrable functions $f:\mathbb{R}\rightarrow \mathbb{C}$ with the usual norm. Let $T\in B(X)$ be defined by $$(Tf)(t)= f(t+1)$$ I need to find the ...
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1answer
32 views

Applications Gelfand-Naimark-Segal Theorem.

I'm reading the book An Introduction to Operator Algebras (By Kehe Zhu). but do not know how to do the following exercise: Let $A$ be the commutative C*-algebra $C(\partial D)$. For any $z\in ...
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Group of operators such that $|T(t)x|\geq c |x|$

Let $X$ be a Banach space. Can I have an example of a strongly continuous group of operators $T(t)$ such that $$|T(t)x|\geq c |x|, \ t\in\mathbb{R}$$with $c>1$. For $c=1$, I know examples of ...
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30 views

norm of a nilpotent matrix

A proof I was reading used the claim that $||{N}||_2$ = 1 for a nilpotent matrix $N$. I tried to prove it, and have a couple of questions on it. First, my "proof": We know that there exists a basis ...
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1answer
33 views

$\sup_t |T(t)|<+\infty$ implies $\sup_t |T(t)^*|<+\infty$?

Let $X$ be a Banach space. $T(t)$ a family of bounded operators for $t\in\mathbb{R}$. $T(t)^*$ is the adjoint operator of $T(t)$. If $\sup_t |T(t)^*|<+\infty$ , then by Hahn-Banach, there's a ...
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2answers
49 views

Numerical Range vs Spectrum

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now. Prove that for normal operators the spectrum is ...
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20 views

Borel Functional Calculus Question

Let $T$ be a bounded operator and $A=\sigma(T)$ be its spectrum. Let $A^n \subset A$ be sequence of subsets s.t $A^n \rightarrow A$ (in compact open topology so $x\in A$ belongs to all but finetly ...
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23 views

Kernel of Integral operator

Let $H: L^2(M) \longrightarrow L^2(M)$ be a bounded operator. Here, $M$ can be a Riemanniannian manifold, or some open subset of $\mathbb{R}^n$. Question: What can I say about the Schwartz Kernel $k$ ...
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1answer
39 views

Adjoint of sum of two operators

Let $A$ be self-adjoint and $B$ symmetric (which means densely defined for me as well) with $A$-bound less than $1$. Does this imply that $(A+iB)^*=A-iB$ ?