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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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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Compact Operators: Decomposition

This is a real question of me. Given a Banach space. Consider a basis on finite dimensional range: $$\dim\mathcal{R}F<\infty:\quad y_1,\ldots, y_N$$ Hahn-Banach lifts the dual basis up: $$ y_n\in ...
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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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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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39 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: Hilbert Spaces [on hold]

Note: This thread is not to gain reputation!! Given a Hilbert space. How to prove: It has the approximation property!
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Approximation Property: Characterization

Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_\varepsilon-1\|_C<\varepsilon\quad(T_\varepsilon\in\mathcal{F}(E))$$ Then every compact ...
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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 an operator algebra. Then a partial isometry satisfies both: ...
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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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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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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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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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Need an operator with given properties [closed]

Need an expression for an operator $U:f\to f^*$ satisfying the following properties: $U$ is linear $U$ depends only on the local properties of $f(x)$ in the neighbourhood of $x=0$ ...
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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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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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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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41 views

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, ...
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Differential operator a bounded operator or not?

Is the operator $T$ a bounded operator mapping $T: H^n([0,\pi]) \rightarrow H^{n-1}([0,\pi])$ ($H^n$ is the n-th Sobolev space with respect to $L^2$) or not? The operator itself is given by ...
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Dynamics: Continuity

Disclaimer: This is a record of results. Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$. Consider a Hamiltonian dynamics: ...
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Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
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Trace class operator

Let $A\in B(H)$ and $\sum_{E}|\langle A e,e\rangle|< \infty$ for every orthonormal basis $E$. Show that $A$ is a trace class (means $\sum_E \langle |A|e,e\rangle < \infty$). I can not prove it. ...
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Explicit analytic continuation of the zeta function and shift operators

Is there a way to compute the radius of convergence of the expansion of the zeta function, e.g. around $a=2+2i$? We have $\zeta(a)=\sum_{n=1}^\infty n^{-a}\approx 0.867.. - i\,0.275.. $, but I ...
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Compact operator and a sot convergent sequence of operators

The following is an exercise of Conway's operator theory: I proved all parts of this exercise except $\|KT_n\| \to 0$. I can easily prove $\|KT_n^*\|\to 0$, but do not have any idea to prove ...
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Show that the operator $(x_n)_n\mapsto (\frac{x_n}{n}) $ is compact

I want to show that the following operator is compact: $$T:\mathbb \ell^p\rightarrow \mathbb \ell^p, \text{ }(x_n)_n\mapsto(\frac{x_n}{n})_n \text{ } 1\leq p<\infty$$ Its the first time that ...
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On Fredholm operator on Hilbert spaces

Let $u: H \to H'$ be a continuous linear operator and $H,H'$ be Hilbert spaces. Let $u^\ast$ denotes its adjoint. By definition, an operator $u$ is called Fredholm if and only if $\ker u$ has finite ...
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Irridicible C*-algebra $A$ implies that projection $p$ is rank one if $pAp=\Bbb C p$

Let $A$ be an irreducible C*-subalgebra of $B(H)$ and $p$ be a nonzero projection in $B(H)$. Suppose $pAp=\Bbb C p$, show that $p$ is rank one. I do not have any idea about it. Please give me a ...
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Spectral measure of a unitary operator

The following is an Exercise of Conway's operator theory: 1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary. 2- Show that every unitary can be so written. 3-Find the ...
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Equivalent finite subspaces of a hilbert space

I have to prove the following statement: Let $H$ be a Hilbertspace and $M,N$ closed subspaces. Then the following holds: If $M \sim N $ and $N$ is finite, then $M$ is finite. I think it should say ...
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Why is $R-\lambda$ invertible for $|\lambda|<1$

I got the following question: Why is $R-\lambda$ invertible for $|\lambda|>1$ and not invertible for $|\lambda|\leq1$ ? R is the right shift operator on $\mathfrak{l^2}$
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$M_{n}(A)$ is an AF-algebra

If $A$ is a $C^*$-algebra that contains an increasing sequence $(A_{n})_{n=1}^{\infty}$ of finite-dimensional $C^*$-subalgebras such that $\cup‎_{n=1}^{\infty} A_{n}$ is dense in $A$, show that ...