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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Showing a bound on a contour integral

I'm working through M. Schechter's 'Principles of Functional Analysis' and I'm working through a proof on page 136 that shows that the spectral radius $r_{\sigma} (T) $ of a bounded linear operator ...
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252 views

When are two commuting linear operators functions of each other

I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up. If we formally consider the integral operators ...
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83 views

Limit of bounded operators

Suppose $T_n$ is a sequence of self-adjoint bounded operators on a Hilbert space, and $T_n \rightarrow T$ in operator norm, $T$ being also bounded and self-adjoint. Do we then have: $T_n^m\rightarrow ...
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55 views

Find a symbol for pseudodifferential operator

$\DeclareMathOperator{\Mel}{M} \newcommand{\Rn}{\mathbb R^n} \newcommand{\dd}{\,\mathrm{d}}$ Consider a pseudodifferential operator (Mellin operator) in positive orthant with symbol $\sigma(z)$: $$ ...
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67 views

Orthogonality & Adjoint Operator

I am trying to prove this simple statement left to the reader in Brézis's book. Let $A \colon D(A)\subset E \longrightarrow F$ be an unbounded operator. Let $G:=\operatorname{Graph}(A)$ and $L=E ...
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291 views

Approximating a Hilbert-Schmidt operator

Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$ where ...
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271 views

Is there an algebra of summable series?

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : \{ Series \} \to \{ numbers \} $ be a regular, linear divergent series operator, which is either one of ...
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97 views

product of bounded linear operators

If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
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90 views

Is there a such thing as an operator of operators in mathematics?

Thus far I have seen operators of numbers and operators that perform on functions like Laplace, Fourier and Z-Transforms but is there an operator in existence that performs on other operators? Like a ...
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443 views

What is the role of supremum in operator norm

An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
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55 views

Prove that the only operator on $\mathbb{C}$ for which his inner product is zero is zero

How do I prove this statement: Let $V$ be an unitary vector space over $\Bbb C$, $(,)$ be an inner product on $V$ and $\Bbb A$ operator $V\rightarrow V$. Then $(\Bbb Av,v)=0$ if and only if $\Bbb ...
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109 views

spectrum of two bounded linear operators

Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.
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177 views

Bounded linear operator in weak topology

Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
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106 views

Is the Strong Limit of a Linear Operator in a Hilbert Space the Same as the Norm Limit?

If $H$ is a Hilbert Space, and I have an operator $F:H \rightarrow H$ which is the limit of a sequence of operators $F_n$ with respect to the operator norm; and this same sequence of operators ...
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368 views

Is the sum of two normal operators normal?

A normal operator is defined as $AA^{*}\ =A^{*}A$ Where A is an operator how do i show the sum of two normal operators is normal? Or find a counter example that shows this is false?
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80 views

Linear functional $\mathscr{L}(E,F)$

Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$. Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question: How to prove ...
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150 views

Proof Involving Difference Operators

Let E be the forward shift operator on $x$ defined by $Ef(x) = f(x+1)$. Similarly, let $\delta$ be the forward difference operator such that $\delta f(x) = f(x+1) - f(x)$ and the inverse operator ...
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225 views

Interchanging closed operators and integrals

I am dealing with a problem in Evans PDE without measure theory knowledge... We have contraction semigroup $\{S_t\}_{t \geq 0}$ on real Banach space $X$, i.e family of bounded linear operators from $ ...
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80 views

How does $\lim A_n$ being not invertible imply $\sup_n\|A_n^{-1}\|=\infty$?

Consider a sequence of operators $\{A_n\}_{n=1}^{\infty}\subset B(X,Y)$, where $X,Y$ are normed vector spaces and $B(X,Y)$ denotes the space of bounded linear operators from $X$ to $Y$. Assume that ...
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196 views

Matrix Representation of Trace Class Operators

Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e: $A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
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232 views

Relation between noncommutative geometry and functional analysis

Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
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How to decompose a representation into direct sum of cyclic representation?

Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
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140 views

How can projection operators be limits of powers of unitary operators?

Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
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884 views

Find norm of the integral operator

Find norm of the following bounded linear operator $$Ax(t)=\int_0^1e^{-ts}x(s)ds$$ where $x\in C[0,1]$ and $t\in[0,1]$. Please help me.
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93 views

Linear operator

Is there a linear bounded (continuous) operator T from $c$ (convergent sequences with sup norm) ONTO $l^1$ (with its usual norm)? If it were so (which seems not), using the open mapping theorem we ...
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59 views

Convergence in norm operator topology

I have to prove that a sequence $A(\varepsilon)$ of operators between Hilbert spaces $A(\varepsilon):H_1\to H_2$ converges, when $\varepsilon\to 0^+$, to an operator $B:H_1\to H_2$ in the uniform norm ...
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64 views

Block Matrices of Operators

I'm trying to prove the following: Consider the vector space of matrices of size $n\times n$ whose entries in $\mathcal B(H)$. Denote this vector space by $M_{n,n}(\mathcal{B(H)})$. We can define ...
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110 views

Counterexample for “the sum of closed operators is closable”

I'm looking for a counterexample in a Banach space. I've seen the counterexample at Sum of Closed Operators Closable?, but I don't understand why $A$ and $B$ are closed. Could someone expand on this ...
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523 views

Calculating the Norm of an operator in $L^2(0,1)$

If I have the following operator for $H=L^2(0,1)$: $$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this: I know that in $L^2(0,1)$ we have that ...
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110 views

Showing an operator is self adjont

I am trying to show that the operator: $$Tf(s)=5s^2\int_0^1t^2f(t)dt+2\int_0^1f(t)dt$$ is self adjoint where $H=L(0,1)$ with real scalars and $t\in \mathcal{L}(H)$. So I can re-write this operator ...
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262 views

Sets $f_n\in A_f$ where $f_{n+1}=f_n \circ S \circ f^{\circ (-1)}_n$ and operator $\alpha(f_n)=f_{n+1}$

Let's start with a function on the Reals (in this case for $x=0$ is not defined): for example $f(x)=b/x$, $x \in \mathbb R$ I define: $$f_0:=f$$ $$f_{n+1}:=f_n \circ S \circ f^{\circ ...
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45 views

Prove that the sequence is in $\ell^{2}$. [duplicate]

Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$ What I've tried so far is ...
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60 views

Symmetric Operator with Different dot products

If I have a symmetric operator $A$ in a metric space $\mathscr{M}$. Then $\langle Au,v\rangle =\langle v,Au\rangle $ with the dot product defined in $\mathscr{M}$. My question is, if I keep the same ...
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41 views

Expectation value of pure state in quantum mechanics

It's well known that in quantum mechanics, the expectation value of a self-adojint operator $A$ in pure state $|\psi\rangle$ is $\langle\psi |A|\psi\rangle = \operatorname{Tr}(A |\psi \rangle ...
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191 views

Hilbert's Inequality

Could you help me to show the following: The operator $$ T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy $$ satisfies $$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p $$ for $1 <p< \infty$ where ...
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573 views

Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces

Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e: $$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
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103 views

The Kernel of unbounded operator in Hilbert space

If $T$ is a densely defined operator from a subspace of a Hilbert space $H$ to a Hilbert space $K$, how to prove that $\mbox{Ker}(T)=\mbox{Ker}(T^*T)$?
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220 views

Unbounded operator $T $ is bounded below when $\overline T$ is bounded

How to prove the following? A densely defined symmetric operator $T$ in Hilbert space $H$ has a closure $\overline T$ which is bounded iff both $T,-T$ are bounded below (there exist constants $c,c' ...
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131 views

Are WOT/SOT topologies hereditarily separable?

Just out of curiosity, Are weak and strong operator topologies on $B(H)$ hereditarily separable? In other words, if $S$ is a subset of $B(H)$, where $H$ is a separable Hilbert space, is $S$ ...
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226 views

Point spectrum in Hilbert spaces

Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
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66 views

A basic result about operators on Hilbert space.

I am studying following result. Let $H$ and $K$ be Hilbert spaces and an operator $A \in B(H, K)$, which has closed range. The spaces $H$ and $K$ have the following orthogonal decompositions: $H = ...
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The Square of the Laplace Transform

I have been looking at the Laplace transform $$\mathcal{L}f(s)=\int_0^{\infty}f(t)e^{-st}dt$$ and I'm trying to find The norm of $\mathcal{L}^2$ The nullspace of The norm of $\mathcal{L}^2$ So ...
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134 views

Strong operator convergence and adjoint operator

Let $H$ be a Hilbert space and $(T_n)_{n \in \mathbb{N}}$ be a sequence of bounded linear operators on $H$. The strong convergence of $T_n$ doesn't imply the strong convergence of $T_n^*$, i.e. ...
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217 views

Invertibility of compact operators

I'm a little confused about compact operators and whether or not they are invertible. Just hoping someone here can clear up my confusion: Let $T$ be a compact operator on a Banach space $X$. Since ...
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1answer
108 views

$K$ is a linear compact operator on Hilbert space $H$. Will the image of $I-K$ on every closed subspace of $H$ be also closed?

Just as the title. We know the image of $I-K$ is closed, but if we restrict $H$ to a closed subspace $V$, will $(I-K)(V)$ be a closed subspace of $H$? Any hint is appreciated.
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105 views

Proof of implication: $\varphi^*\text{ is bounded below}\implies\varphi\text{ is a quotient map}$

We say that a bounded operator $\varphi:X\to Y$ is $c$-topologically injective if $\Vert\varphi(x)\Vert\geq c\Vert x\Vert$ for all $x\in X$ $c$-topologically surjective if for all $y\in Y$ there ...
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217 views

Composition of pseudo-differential operators

Denote by $S^m$ the set of functions $p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that $p \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and $$\left| ...
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440 views

Square root is operator monotone

This is a fact I've used a lot, but how would one actually prove this statement? Paraphrased: given two positive operators $X, Y \geq 0$, how can you show that $X^2 \leq Y^2 \Rightarrow X \leq Y$ (or ...
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221 views

Inverse of Identity plus Volterra operator

consider the following operator or $L_2(0,1)$, $(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial. I am trying to construct the inverse of this ...
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122 views

does invertibility of product imply invertibility of each term of product?

Suppose $\mathcal{H}$ is a Hilbert space and the product $T_1T_2 \in B(\mathcal{H})$ is invertible. Does this imply that both $T_1, T_2$ are invertible ? I am unable to prove this since, unlike the ...