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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Adjoint operator of $L^\infty$

Lets denote with $(\Omega,\Sigma,\mu)$ a $\sigma$-finite measurble space with a linear, continuous operator $$T : L^\infty \to L^\infty.$$ Does this always imply the existence of a linear, continuous ...
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Symmetries on Hilbert spaces

Let $\mathfrak{H}$ be a Hilbert space and let $\mathcal{E}(\mathfrak{H})$ be the set of all operators $T\in B(\mathfrak{H})$ such that $0\leq T\leq 1$ (these operators are also called effects on ...
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Essential self-adjointness of the Laplace operator via the Fourier transform

I'm working through some notes on showing the essential self-adjointness of the Laplace operator on $\Bbb R$ via the Fourier transform (see here) but there seems to be a little bit of liberty taken at ...
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Riemann manifold with unbounded Laplacian

How can one characterize a Riemann manifold the Laplacian of which is unbounded? (Equivalently, what are those manifolds on which the Laplacian is bounded? I am interested in working with its ...
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59 views

Is the injectivity of the operator equivalent to the surjectivity of its adjoint

Let $X$ and $Y$ be two normed linear spaces. Let $T:X \to Y^*$ be a linear operator (not necessarily continuous) and let $T^*$ be its adjoint, i.e. $T^*:Y \to X^*$ is defined by $ \langle T^*y,x ...
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Existence of a linear operator

Does there exist a linear operator $A:\ell^2 \to \ell^2$, $$ A(x_1, x_2, \ldots, x_n, \ldots) = (y_1, y_2, \ldots, y_n, \ldots) $$ such that $\exists x = (x_1, x_2, \ldots, x_n, \ldots)$: ...
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27 views

Compact space X is totally disconnected if and only if C(X) is generated by its projections

If $X$ is compact, show that $X$ is totally disconnected if and only if $C(X)$ as a C*-algebra is generated by its projections. My attempt: Suppose $X$ is totally disconnected, then $X=\{x_i\}_{i\in ...
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38 views

Linear operator norm

I am trying to show that these two definitions for a bounded linear operator norm on the normed linear space $X$ are equivalent: $$ \sup\{T(x)\,:\, \|x\|\le 1\}=\|T\|_*=\inf\{M>0\,:\, T(x)\le ...
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1answer
24 views

A non-continuous idempotent linear operator in a Banach space

Does there exist a non-continuous idempotent linear operator $P: X \to X$ where $X$ is an infinite-dimensional Banach space? That is, $P^2 = P$, and there is a sequence $\{x_n\}$ of elements of X such ...
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33 views

Orthogonal projectors on non-orthogonal subspaces

It is a well known fact that if(f) $V,W$ are orthogonal subspaces of a Hilbert space $H$, then their orthogonal projectors satisfy: $$ P_{VW} = P_V + P_W, $$ where $P_{VW}$ is the projector on $V+W$. ...
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Strong resolvent convergence and spectral measures

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in the strong resolvent sense. Denoting by $E_n$ and $E$ ...
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Discrete bi-Laplacian

I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges. In particular ...
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References on projectors

What are good books or articles about linear projectors in Hilbert spaces? I am mostly interested in the finite dimensional case (but anything is welcome). All about idempotents, orthogonal and ...
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60 views

Perturbation of Laplacian

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; ...
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39 views

Proving equivalence of two definitions

Hi all I am intersted in proving the equivalence of the following two definitions of pseudomontoncity: Let $V$ be a reflexive Banach space and $K \subset V$ closed and convex. Definition 1: $A: V ...
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1answer
30 views

the sum of two unbounded normal operators

why A and B are normal?and why "0" is not closed on H1(R)?
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36 views

Approximation Property: Decomposition

This is a real question of me. Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional range: ...
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Question about linear operator continuity

Let $A:X\rightarrow Y$ be a linear operator, $X,Y$ normed spaces. Show that a linear operator is continuous (bonded) if for every sequence $x_n\rightarrow 0$ in $X$ has a bounded image $Ax_n$ in $Y$. ...
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35 views

Does this operator equation have solutions?

Hi Math StackExchange community, I have a question that originates from a Physics problem; the question itself however is about solving an operator equation. In a particular quantum mechanical ...
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45 views

How do I prove this statement about the operator norm?

I stumbled across this equation in a paper, which may seem obvious, but I'm wondering if someone can explain why this is true? By definition of an operator norm, $$\left[(D^*D)^{-1} - ...
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Approximation Property: Characterization

As reference the german wiki: Approximationseigenschaft Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$ ...
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40 views

Inverse operator of $I-A$

Let $H$ be an Hilbert space, $A:H\to H$ be a bounded linear operator such that $$ \|A^{n_0}\|< 1\qquad\text{for some}\quad\; n_0\in\mathbb{N}. $$ I have to show that $I-A$ is invertible. My idea ...
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Partial Isometries: Subspaces

Note: This thread is not to gain reputation!!! Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a bounded operator: $$W:\mathcal{H}\to\mathcal{K}:\quad\|W\|<\infty$$ Then a partial ...
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Adjoints of operators between different Hilbert spaces.

When we have an operator $$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$ from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint ...
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Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to ...
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Separating and cyclic vector

Let $\{\Gamma_i , \mu_i\}_{i\in I}$ be a family of probability measure spaces and suppose $I$ is uncountable. Let $\{\Gamma , \mu\} = \prod_{i\in I} \{\Gamma_i,\mu_i\}$ be the product measure space. ...
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24 views

Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank ...
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Is this a bounded linear map?

I tried very hard to (dis)prove it, but now I give up. Define a map which maps $x\in L_2[0,1]$ to the function $$(Tx)(t) = \frac{1}{\sqrt{t}}\int_0^t \frac{x(s)}{\sqrt{s}} \,d s.$$ I don't even ...
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Representation of linear functional on $c$

On the space of convergent sequences $c$ let $x=(x_i)_{i\in \Bbb N}\in c$ and $\lim_{i \to \infty}x_i=x_0$ then a bonded linear functional on $c$ has a representation ...
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20 views

Action of projections

Suppose we have a projection $p$ on a Hilbert space $\mathfrak{H}$. Is the following true: There exists an set $V\subset\mathfrak{H}$ such that $p(x)=x$ if $x\in V$ and zero else? I asked because I ...
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Orthogonality of projections on a Hilbert space

Assume that $p$ and $q$ are (orthogonal) projections on Hilbert space $\mathcal{H}$. I want to prove: $pq=0$ iff $p+q\leq1$ I had the following in mind: Assume $pq=0$. Then $qp=0$, hence $p+q$ is a ...
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1answer
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Image of bounded linear operator?

Let $x^\ast$ be a continuous linear functionals on $l_p$. Let $(e_i)_{i\in \Bbb N}$ be the standard basis of $l_p$. Consider $y=(y_i)_{i\in \Bbb N}$ the sequence defined by $y_i=x^\ast(e_i)$. Let ...
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Elliptic differential operator

I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are ...
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Is this a bounded linear operator?

I have the following problem. Show that $y_n={1 \over \sqrt n}\int_0^1t^nx(t)dt$ is a bounded linear operator that maps $L_2[0,1]$ into $l_2$ with the usual norm on the respective spaces. My approach ...
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Algebraic multiplicity of an eigenvalue for abstract operators

How does one define algebraic multiplicity of an eigenvalue for an abstract operator? (for a matrix the definition is clear). E.g. Consider $\partial_x^2$ on $H^2_{per}(0,1)$ then $\partial_x^2 ...
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1answer
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spectrum of convolution integral operator

Let $A f(x)= \int_{-\pi}^{\pi} h(x-y) f(y) dy$ operator $L^2( {-\pi},{\pi})->L^2( {-\pi},{\pi}), h$ is continuous, periodic with period $2\pi$ and $h(x)=h(-x)$ on $ [ {-\pi},{\pi}] $. How can I ...
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Self-adjoint operator- domain unique?

I was wondering about the following: Let $T : dom(T) \subset H \rightarrow H$ be a self-adjoint operator, does this mean that the domain of $T$ is uniquely defined or is it possible to make the same ...
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characterizing an operator with projection whose spectrum is contained in $\{-1,1\}$

Let $\mathcal{A}$ be a $C^{*}$-algebra and $\sigma$ denote the spectrum. I want to show that if $\sigma (A)\subseteq \{-1,+1\}$ for $A\in \mathcal{A}$ then there is a projection $P$ such that ...
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Reference for unbounded operators

I've run into some unbounded operators in my research and need to learn some of the theory of unbounded operators. Particularly I want a rigorous treatment that discusses symmetric operators, ...
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Pseudodifferential Operators are properly supported iff the symbol is properly supported

I am trying to proof the following statement: Pseudodifferential Operators are properly supported iff the symbol is properly supported. A Pseudodifferential Operator $A \in \Psi^m(X)$ ($\Psi^m$ ...
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The domain of the generator of a strongly continuous semi-group is dense

Let $A$ with domain $D(A)\subset X$, ($X$ Banach) be the generator of a strongly continuous semi-group $(S(t))_{t\geq 0}$. Then $D(A)$ is dense in X. I am not sure if this proof is correct. I know ...
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1answer
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Using substituion rule for piecewise monotonic function

I am trying to read the ON THE EXISTENCE OF INVARIANT MEASURES FOR PIECEWISE MONOTONIC TRANSFORMATIONS paper from 1973. One has given a map $\tau : [0,1] \to [0,1]$ which is a piecwiese monotonic ...
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Adjoint of Integral Operator in $L^p$

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Find the adjoint of $T$. I know how to this in the case $p=2$ as shown here. But in general $L^p$ is not an ...
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Does every closed, densely operator in a Banach space have an closed, densely defined extension on a Hilbert space?

Assume that $H_1,H_2$ are separable Hilbert spaces, $B$ is a separable Banach space and $H_1\subset B\subset H_2$. Assume further that the inclusion mappings are continuous and have dense images. ...
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Møller Operators: Summary

Disclaimer This thread is meant as summary. For more informations see: SE blog: Answer own Question MSE meta: Answer own Question (The second especially reveals the opinion of the community!) ...
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Bound for Integrator Operator

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Prove that $T$ is compact on $E$. I would like to use Ascoli-Arzela', but I need to prove: $$|T u(x) − T u(y)| ...
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Boundedness of linear operators in $L^p$

I was wondering if the following conditions are equivalent for a linear operator $T$ between $L^p$ spaces: i) T maps $L^p$ to $L^p$, that is, if $f \in L^p$ then $Tf \in L^p$. ii) There exists ...
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Hilbert- Schmidt class is an ideal

Definitions: 1 - An operator $y\in B(H)$ is said to be of trace class if $y$ is compact, and also $\sum|\alpha_n| <\infty$ where $\alpha_n \in \sigma(y)$ and $y$ has a representation $\sum ...
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weak convergence of $L^2$ implies weak convergence of $W_0^{1,2}$ (up to a subsequence)?

In the paper that I am reading, it says that if $\{u_n\}$ are bounded in $W_0^{1,2} (\Omega)$ (bounded $\Omega\subset \mathbb{R}^N$) and $u_n \rightharpoonup u$ weakly in $L^2 (\Omega)$, then there ...
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Compact operator space is the greatest ideal of $B(H)$

Suppose $H$ is a separable infinite dimensional Hilbert space. Show that if $A\in B(H)$ is noncompact, then there exist two operators $B,C$ such that $BAC=1$. Clearly if $A$ is invertible it holds, ...