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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Sum of the matrix series

Let $A\in\mathbb R^{n\times n}$ be a symmetric matrix which $0\preceq A\preceq I$ ($I$ is identity matrix), and $w_k\in\mathbb R^n$ are arbitrary certain vectors which $\|w_k\|\leq1,\,\,k=0,1,\ldots$ ...
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1answer
26 views

Summation of the Bergman kernel at two distinct points is constant?

Let $\Omega$ be a bounded simply connected domain in $\mathbb{C}.$ Let $K(z,w)$ denotes the Bergman kernel of $\Omega.$ Let $w_1,\,w_2$ be two distinct points in $\Omega.$ I'm looking for a domain ...
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0answers
79 views

Troublesome proof in Functional Analysis with dual vector space

Greetings to all of you I have tried to prove the following theorem but I am having some troubles with it. Let $X$ be a separable normed space and $(x_n')$ a bounded sequence in $X'$, then there is a ...
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1answer
171 views

Spectrum of shift-operator

Hoi, consider the Hilbertspace $l^2$ and the Left and Right-shift operator \begin{align*} L(x_1,x_2,\cdots) &= (x_2,x_3,\cdots)\\ R(x_1,x_2,\cdots) &= (0,x_1,x_2,\cdots ) \end{align*} I know ...
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13 views

Subordinate operators

Let $A$ be a linear densely defined operator on a Hilbert space $H$ and $L$ is a selfadjoint operator with discrete spectrum such that $\mathcal{D}(L) = \mathcal{D}(L)$ and $$\|Tf\| \leq M ...
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60 views

Conjugate convex functions property

$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$ And I ...
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78 views

Ideals in $B(H)$ are self-adjoint

It is known that every (closed two-sided) ideal in a $C^{*}$-algebra is self-adjoint. The proofs that I've seen involve functional calculus and approximate units. I am wondering whether there is a ...
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35 views

A question on a sequence in a Banach algebra [duplicate]

If $\{u_{k}\}_{k=1}^{\infty}$ is a sequence in an Banach algebra (and more specifically, in the set of all the bounded linear operators of a Banach space $X$). If ...
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1answer
61 views

Operator's norm

Let $T$ be a linear densely defined operator on a Hilbert space $H$ and $L$ be a selfadjoint operator with discrete spectrum and $T^{-1}$ is bounded such that $$\|Tf\| \leq M \|Lf\|^{a}\|f\|^{1-a}, ...
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81 views

A question on the spectral projection

I am reading a paper about spectral theory. And I meet with some problems. An operator $K\in L(X)$ is said to be algebraic if there exists a non-trivial complex polynomial $h$ such that $h(K)=0$. By ...
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1answer
110 views

Prove that if transformation matrix is unitary, then the basis is orthonormal

V is a vector space with the complex field, B is an orthonormal basis of V , and C is some arbitrary basis. Prove that if the transformation matrix from basis C to B is unitary, then C is also ...
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57 views

Explicit operator in separable Hilbert space

This is a question about (possible unbounded) operators. We know that $\mathcal{D}(T^*)=\{0\}$ iff $\mathcal{G}(T)$ is dense in $\mathcal{H}\times\mathcal{H}$, where $\mathcal{H}$ is a separable ...
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74 views

Normal compact operator commute with bounded self adjoint operator in Hilbert space.

Suppose $H$ is a Hilbert space and $A:H\rightarrow H$ is a normal compact operator such that $\ker(A)=0$. show that if $B$ is a bounded self adjoint operator that commutes with $A$ then the spaces in ...
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2answers
64 views

Clarifying the definition of essential self-adjointness

If a Hilbert space operator $T$ has a unique self-adjoint extension, must the extension be the closure of $T$?
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generalizations of continous operators?

What are generalizations of the notion of continous linear operator $P:X\to X$, where X is a Banach space? I'm looking for some broader class of operators that nevertheless share some properties of ...
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42 views

A simple question about completely positive linear maps

Let $A$ be the C*-algebra and $M_{n}(A)$ be the C*-algebra of $n\times n$ matrices with entries in $A$. We use $(a_{ij})$ to denote the element of $M_{n}(A)$. My question is: For every $a\in A$, ...
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49 views

Half Laplace operator

I'm curious whether a half Laplacian (or square root of Laplacian) exists. More specifically, I'm looking for an $X:C^2(\Bbb R^n)\to C^2(\Bbb R^n)$ operator such that $$\forall f:XXf=\Delta f$$ I know ...
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Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
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1answer
40 views

How to prove the demicountinuity of nonlinear operators?

Define a nonlinear operator $\mathbf{J}(\mathbf{x}):~\mathbb{R}^3 \rightarrow \mathbb{R}^3$ as $$ \mathbf{J}(\mathbf{x}):= |\mathbf{x}|^{-\alpha}\mathbf{x},~0<\alpha<1. $$ How to prove that ...
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86 views

Relationship between adjoint operators, trace-class operators, compact operators and density operators in Quantum-Mechanics

I don't know much about Functional Analysis, but I was wondering about the following: In Banach spaces it is possible to define for every continuous opertor $T:X \rightarrow Y$ an adjoint Operator ...
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48 views

Complex Power of a differential operator

Let $(X,\|\cdot\|)$ be a Banach space and consider a sequence $B_n \colon X \to X$ of bounded operators. I remember from my course in operator theory that the partial sum $$ S_N = \sum^N_{n = 1} B_n ...
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38 views

Example of Hilbert space operator that is not a product of unitary and positive

If $A$ is a unital $C^{*}$-algebra, and $a\in A$ is invertible, then $a=u|a|$ where $u$ is unitary and $|a|=(a^{*}a)^{1/2}$ is positive. I am looking for an example of a bounded linear operator on ...
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28 views

Show, that $T\colon C([a,b])\to C([a,b])$

I have a question concerning an integral equation that is written as an fixed point equation, namely $$ u(x)+\int_a^x F(x,y,u(y))\, dy=f(x,u(x)),~~x\in [a,b] $$ with $$ ...
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71 views

When an invertible element in a $C^{*}$-algebra is unitary

I am trying to show that if $a$ is an invertible element of a unital $C^{*}$-algebra, and $||a||=||a^{-1}||=1$, then $a$ is unitary. I can do this if I think of $a$ as a Hilbert space operator using ...
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84 views

Using Nemytskii Theorem for Sobolev Spaces

The Nemytskii mappings in Lebesgue spaces theorem is as follows: If $a: \Omega \times \mathbb{R}^{m_{1}} \times \cdots \times\mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is a Caratheodory ...
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2answers
87 views

Functional Analysis, operator theory, eigenvalues of a operator

We have $$T_\alpha:C[a,b]\to C[a,b]$$ $$T_\alpha f= \alpha f$$ where $C[a,b]=\{ f:[a,b]\to \mathbb{R} \quad f$ is continuous} and $\alpha\in C[a,b]$ fixed. Show: Spectrum of $T_\alpha\equiv ...
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105 views

Question about projections on Hilbert space

Let $P_i$ be projections from a Hilbert space $\cal{H}$ to its closed subspace $\cal{H}_i$, $i=1,2,\cdots,n$, such that $\sum^n_{i=1} P_i$ is also a projection. And let $P$ be a projection from ...
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1answer
60 views

Find the norm of $A$ where $(Af)(t)=tf(t)$

I have the following problem that I would like to ask you about: I have $X$ as my normed linear vector space and $B(X,X)=B(X)$ as my space of all operators $A: X \to X$, where for all $A \in B(X)$ is ...
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Uniqueness of Unitary operator

i saw the post "Polar decomposition normal operator" (Polar decomposition normal operator). There was that such a $U$ is unique iff the image of $T$ is dense. Some lines later by the comments there is ...
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40 views

Question about domains of unbounded operators

This is a part of a theorem in Rudin's Functional Analysis, in the chapter on unbounded operators. Let $\mathcal M$ be a $\sigma$-algebra in a set $\Omega$, $H$, a Hilbert space and $E:\mathcal ...
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55 views

determinant identity for invertible finite rank operators

I am currently reading a paper where the following identity, valid for an invertible finite - rank operator $T \colon \mathscr{H} \to \mathscr{H}$ on a separable Hilbert space, is given: $$ \log \det ...
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51 views

Three basic questions about linear operator in a Hilbert space

Just come across three questions in reading a paper. Suppose we are dealing with a Hilbert space of $L_{2}[0,1]$ and all the functions mentioned below are in $L_{2}[0,1]$. Define the operator $A$ by ...
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Finding an isometry between two subspaces of a Hilbert space

So, I'm given a Hilbert space which is the direct sum $H=H_1\oplus H_2$ of two separable Hilbert spaces $H_j$. There is a closed subspace $D\subseteq H$ which satisfies that it is not a subspace of ...
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51 views

Interpretation of Fredholm Alternative with respect to PDEs

I have studied the Fredholm Alternative, which states the following: Theorem: Let $K:H \rightarrow H$ be a compact linear operator and let $I$ be the identity operator on $H$. Then: 1.$N(I-K)$ is ...
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45 views

Minkowski functional and strange theorem

I have a theorem that says the following: Let X be a normed space and $U\subset X$ a convx subset with $0 \in \text{int(U)}$, then we have: $U$ is absorbing and if $\{x;||x|| < \epsilon\} \subset ...
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Fredholm alternative and orthonormal basis

The following question relates to the Fredholm alternative: Let $K:H \rightarrow H$ be a compact linear operator and let $I$ be the identity operator. Notation: $N$ is the nullspace and $R$ is the ...
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139 views

Norm of a matrix equals greatest eigenvalue

How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an ...
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1answer
37 views

Pulling Operator Inside Integral

Say $Y$ is a Banach space and you have a family of continuous/bounded operators $L_{x}: Y \rightarrow Y$ for $x\in \mathbb{R}$ and say you have an bounded, smooth map $f(x):\mathbb{R}\rightarrow Y$. ...
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40 views

Predual of $\mathcal{B}(K, H)$

Is there a predual of $\mathcal{B}(K, H)$? So, what does the space $X$ look like, such that $X^*=\mathcal{B}(K, H)$.
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If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)

Let $V,W$ be normed vector spaces, and $L(V,W)$ be the space of bounded linear operators. Usually I would only see the statement "If $W$ is Banach, then $L(V,W)$ is Banach.". But Wikipedia writes that ...
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1answer
40 views

questions about norm of integral operator

The following is a question I came up with when I was studying the same problem in dimension 1 (for which also I have the questions that follows) but I put in generality. Let $U_1, U_2 \subset ...
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Find the spectrum of this operator

$$O:l_p\to l_p\quad p\in[1,\infty]$$ $$Ox=(0,x_1,x_2,x_3,...)\quad \forall x=(x_1,x_2,x_3,...)\in l_p$$ if $\lambda\ne0$ then we have $$\lambda x-Ox=(\lambda x_1,\lambda x_2-x_1,\lambda ...
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Integral kernel of the resolvent operator

Suppose we have an explicit formula for the integral kernel $k(x,y)$ of an operator $D$ acting on smooth $\mathbb{C}^n$-valued functions defined on an interval $[0,\beta]$, that is $$ Df(x) = ...
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Why do $S^{1/2}$ and $T^{1/2}$ commute

This question is actualy related to my old question Product and sum of positive operators is positive If $S,T \in B(H)$ are bounded, linear and normal operators on a Hilberspace $H$, i.e. $SS^*=S^*S$ ...
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42 views

Prove this sequence in $L(\ell_1)$ does not converge to zero?

I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow ...
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88 views

difference between idempotent and projection operators

in book of conway, functional analysis, section operators on Hilbert space(projection and idempotent) say that a projection is an idempotent such P that $(kerP)=(rangP)^\perp$. but from the next ...
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Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on ...
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$\gamma-$radonifying operators.

I am reading about $\gamma$-radonifying operators, and came across 2 similar definitions. And I hoped to understand why they are equivalent. Let $H$ be a seperable real Hilbertspace, $E$ banach ...
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56 views

Positive unbounded operator with zero not as an eigenalue

I am currently doing Quantum Mechanics and I am supposed to show that zero is an eigenvalue of a positive operator. I have no knowledge of Functional Analysis at that kind of level, so I was wondering ...
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1answer
60 views

completely continuous implies compact

I'm searching for a proof of the fact that if: $T$ is a bounded operator in a reflexive Banach space that maps weakly convergent sequences onto convergent sequences then $T$ is compact. If we let ...