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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Finding this operator's spectrum

In an exam, my professor gave the following exercise: State and prove the spectral theorem for compact operators. Let $K$ be the operator defined by: $$Kf(t)=\int_0^1\min(t,s)f(s)\mathrm{d}s.$...
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Quotient map and compactness

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. Let $\pi :X\longrightarrow \frac{X}{Y}$ be the quotient map. If $\pi T\arrowvert _Y$ be a compact operator, can we say there exists a ...
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17 views

To be closed in weak operator topology

I know the set of finite rank operators with rank less than n is closed in strong operator topology. Can we say it is also closed in weak operator topology?
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If $A$ is a compact operator, is $\overline{A(B_1(0))}$ finite dimensional?

Let $A$ be an operator. An operator is called compact iff $\overline{A(B_1(0))}$ is compact. A normed space is finite iff $\overline{B_1(0)}$ is compact. Let $X$ be a Banach space and $Y$ a Hilbert ...
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1answer
30 views

Is this Adjoint Operator Self Adjoint?

I'm helping some students study for their qualifying exam, and I wanted to double check my interpretation of a question. Suppose we define the operator $L$ on $H=L^2\left([0,\infty)\right)$ so that ...
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Characterization of the Gradient of a Distribution

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ (without topology) $u:\mathcal D(\Omega)\to\mathbb R$ is called distribution on $\Omega$ $:\...
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1answer
25 views

Is $p \vee q \leq p+q$ for $p,q$ projections?

I am wondering if $p \vee q \leq p+q$ for $p,q$ projections acting on some Hilbert space $H$. In particular, I wonder if the set of finite trace projections is upwards directed with the usual ...
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1answer
42 views

All nonzero singular values of $A$ are equal to $1$ iff $A^*=A^*AA^*$ and $A=AA^*A$

I want to show that all the non-zero s-numbers, i.e. singular values $s_j(A):=(\lambda_j(A^*A))^{1/2}$, of A (a bounded linear operator of finite rank acting on a separable Hilbert space $H$) are ...
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40 views

Show that $A\varphi_j=\left<A\varphi_j,\varphi_j\right>\varphi_j$ and $A^*A\varphi_j=s_j(A)^2\varphi_j$ for all $j$

Let $A$ be a bounded linear (compact) operator acting on a separable Hilbert space $H$, and let $\varphi_1,\varphi_2,\ldots$ be an orthonormal basis of $H$. I Assume that $|\left< A\varphi_j,\...
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How can we extend $\operatorname{div}:C_c^\infty(\Omega)^d\to L^p(\Omega)$ to $W_0^{1,\:p}(\Omega)^d$?

Let $d\in\mathbb N$ and $p\ge 1$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open and $$W_0^{1,\:p}(\Omega):=\overline{C_c^\infty(\Omega)}^{\left\|\;\cdot\;\...
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41 views

The form of a normal operator with only one element in its spectrum

Let be $H$ a Hilbert space. Show that if $T$ is a normal linear operator continuous (i.e. $T^*T = TT^*$, with $T^*$ the Hilbert adjunct of $T$) and your spectrum $\sigma(T) = \{\lambda\}$, than $T = \...
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1answer
71 views

What is the mathematical meaning of a quantum operator?

(Context: I am learning functional analysis using the book by Erwin Kreyszig "Introductory functional analysis with applications." The last chapter is dedicated to the applications of functional ...
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17 views

Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \begin{equation} \vert \vert T \vert \...
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26 views

On sharp bounds of some dyadic operators

I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. For example, the martingale transform $T_{\sigma}$ which ...
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37 views

How to find operator with Fibonacci eigenvalues?

How can I find the operator that satisfies this equation? $$F_nx^n=Dx^n$$ Summing over $n$ we can rewrite this as $$\frac1{1-x-x^2}=D\frac1{1-x}$$ I am unsure whether this can be solved. I am ...
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1answer
12 views

Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
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1answer
22 views

Countable weighted shift has no invariant subspace.

Suppose I have $T(e_n)=w_ne_{n+1}$ where $w_n>0$ (and are bounded) and $\{e_n\}$ denotes the canonical basis of $l^{2}(\mathbb{N})$. I would like to prove that the only (closed) invariant ...
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59 views

Exponential of Operators

Let $H$ be an Hilbert Space $\exp(T)$ the exponential for an operator $T \in L(H)$. I know that $\exp(A)^{*} \exp(A)=\exp(A) \exp(A)^{*}=id$. Can I conclude that $A^{*}A=AA^{*}$? Cannot find an ...
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37 views

If $Tv=\mu v$ for some $\mu>0$, then $v\in\ker(T^{1/2})^\perp$

Let $V$ be a separable $\mathbb R$-Hilbert space $T$ be a bounded, linear, nonnegative and symmetric operator on $V$ $(v_n)_{n\in\mathbb N}$ be an orthonormal basis of $V$ with $$Tv_n=\mu_nv_n\;\;\;\...
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33 views

If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
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1answer
39 views

A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$

As the title says, the question is how to prove $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$
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$S$ is continuous with Weak * topology from $Y^*$ to $X^*$ if an only if $S=T^*$ for some $B(X,Y)$ [duplicate]

How to prove that prove that $S$ is weak$^*$-continuous from $Y^*$ to $X^*$ if an only if $S=T^*$ for some $T\in B(X,Y)$ Thanks for any hints. To show that $T$ is continuous is straight forward ...
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1answer
37 views

What is the relation between the matrix of an operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, either both real or both complex, and let $T \colon X \to Y$ be a linear operator. (Then $T$ is bounded since its domain is finite-dimensional). ...
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Prob. 5, Sec. 4.5 in Kreyszig's functional book: The adjoint of the composite of two bounded linear operators

Let $X$, $Y$, and $Z$ be normed spaces, either all real or all complex. Let $T \colon X \to Y$ and $S \colon Y \to Z$ be bounded linear operators. Let $X^\prime$, $Y^\prime$, and $Z^\prime$ denote the ...
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61 views

Given $Q:ℝ^d→(\text{Hilbert-Schmidt operators }U→ℝ^d)$, find a Hilbert-Schmidt operator $T:U→L^2(ℝ^d,ℝ^d)$ with $Q(x)u=(Tu)(x)$

Let$^1$ $U$ be a separable $\mathbb R$-Hilbert space $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be a bounded domain $H:=L^2(\Omega,\mathbb R^...
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How to prove $n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$ is a norm on $B(H)$ and $n(T)\lt\|T\|\lt2n(T)$ where $T\in B(H)$? [closed]

Let $H$ be a Hilbert space over $\mathbb C$. If $T\in B(H)$, how to prove that $$n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$$ is a norm on $B(H)$ and $$n(T)\lt||T||\lt2n(T)\ \textrm{?}$$ I couldn'...
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1answer
20 views

Applications of Positive Operator Valued Measures (POVMs)

I am wondering what some applications of POVMs are in mathematics (or mathematical physics)? I am going through Berberian's 'Notes on Spectral Theory', which shows how we can write a normal operator ...
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88 views

Restriction of operators on $l_\infty$ to $c_0$

Given $\epsilon>0$, can we always find a non-compact operator $T:l_\infty\to l_\infty$ of norm larger than $1$ such that the restriction of $T$ to $c_0$ is compact and has norm smaller than $\...
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1answer
19 views

Continuous inverse of an unbounded operator on a Hilbert space

Let $T:D(T)\to H$ be an unbounded densely defined operator on a Hilbert space $H$. Suppose that $T^{-1}$ is continuous, i.e. that $0$ belongs to the resolvent set $\rho(T)$ of $T$. As $T^{-1}$ exists,...
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1answer
45 views

Norms are equivalent iff dual spaces for them are the same?

It is trivial that if we have a vector space $X$ and two equivalent norms on it than $X'_1$ -dual space (of continuous functionals) for the first norm and $X'_2$ are the same spaces. Is the converse ...
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19 views

Proving that $-\Delta+V$ on some domain is self-adjoint

This question may look as a "proof-reading" question, but what I ask is if I correctly understand the way these concepts work, by showing how I think about them. Suppose I have the following three ...
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1answer
36 views

operator inequality using spectral theorem

Given two densely defined unbounded self-adjoint strictly positive operator $A$ and $L$ in Hilbert space $H$ with domain $D(A) \subset D(L)$ and $\|Lx\| \leq \|Ax\|$ for all $x\in D(A)$, why do we ...
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29 views

Boundedness of a naive integral operator

Define integral operator $J:L^2[0,1] \to L^2[0,1]$, $$Jf(x) := \int_0^x f(s) ds.$$ I am wondering if the following equivalence holds, $\|Jf\|_{L^2} \simeq \|f\|_{H^{-1}}$, where $\|\cdot\|_{H^{-1}}$ ...
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1answer
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A restriction $A_2$ of a compact self-adjoint compact linear operator $A$ is also compact and self-adjoint?

Let $X$ be an inner product space and let $A$ be a compact and self-adjoint linear operator. Let $p_1$ be an eigenvector of $A$. Let $A_2$ be the restriction of $A$ to $X_2$ where $X_2$ is given by $$...
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1answer
26 views

Question on spectral theorem for compact operators

I'm studying a proof of the spectral theorem for compact operators. The first part of it reads as follows: Let $X$ be an infinite dimensional inner product space and let $A: X \to X$ be a compact and ...
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1answer
23 views

Operator commutes with spectral projection

Let $E$ be the spectral measure to an (unbounded) self-adjoint operator $A$. Is there a sufficient and necessary condition so that for a bounded interval $I$ we have $E_I A= AE_I$?
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Proof of matrixnorm relative to one-norm

Let $A \in \mathbb{R}^{d \times d}$ a $d \times d$-matrix $A=(a_{ij})$ with norm $\|\cdot\|_1$. Proof: $$\|A\|= \max_\limits{j=1,...,d} \sum_\limits{i=1}^d |a_{ij}|$$ Let $\|x\|_1=1$ and $Ax=y$: $\|...
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1answer
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what is the reason of this of the following statement?

in a paper i saw the following statement: Let $\Phi:B(X)\longrightarrow B(X)$ is an additive and surjective map. If $T\in B(X)$ and for some $x\in X$ $Tx \otimes {\Phi(T)}^*f=\Phi(T)x\otimes T^*f$ ...
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Prove for the family of operators $\{S(t)\}_{t>0}$ that $S'(0)$ exists

Let's consider a real-valued function $V\in L^\infty(\mathbb{R})$ and a family of operators $\{S(t)\}_{t>0}$ defined on $L^2(\mathbb{R})$ as follows: $$\qquad \qquad\left(S(t)f \right)(x) = \frac{...
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1answer
37 views

Is this a general identity for the Resolvent? [solved: integral representation of the resolvent]

And if so, what is it called? $$i(H-\lambda - i\epsilon)^{-1}\phi = \int_0^\infty e^{-\epsilon t} e^{i\lambda t}e^{-iHt}\phi\,\text dt$$ as in Reed-Simon XIII.7 example 1. It is stated there for $H=-i\...
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74 views

Integrating a Linear Operator $A:H\longrightarrow H$ (Matrix)

I am trying to prove a functional analysis proposition, but I got stuck. I have to integrate a matrix. In my proof I use the following matrix: Let $A$ be a self-adjoint matrix on $H=\mathbb{C}^n$ ...
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1answer
26 views

Non-self Adjoint Operator Algebra References

The problem I am working on has led me to define a norm closed sub-algebra $\mathscr{A}$ of $\mathscr{B}(\mathscr{H})$. The algebra is generated by some mild relations, and in general, will not be ...
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38 views

Injective Integral Operator on $L^2[0,1]$ or $C[0,1]$?

Consider an arbitrary $f \in L^2 [0,1]^+ $ where $L^2[0,1]^+$ is the function space of square integrable non negative functions. We say $T$ is an Integral Operator if $T$ is of the form , $$ T(f) = ...
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Calculate the trace of $LBB^*$, where $L:H→H$ and $B:=ΦT^{1/2}$ for some $Φ:U_0→H$, an embedding $ι:U_0→V$ and $T:=ιι^*$

Let$^1$ $U$, $V$ and $H$ be $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\...
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Application of Uniform boundedness theorem: $\langle Tx,y\rangle$ bounded for each $x,y$ then $||T||$ is bounded

For Hilbert Space $X$, if we have a condition on a subset $F\subset B(X)$ ('set of bounded linear operators on $X$') such that $$ \{\langle Tx,y\rangle:T\in F\} $$ is a bounded set for each $x,y\in ...
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25 views

Eigenvalues of Finite Type

I want to show that the following holds: Let $T:X\rightarrow Y$ and $S:Y\rightarrow X$ be operators acting between Banach spaces. Assume that $\mu \not=0$ is an eigenvalue of finte type of $ST$. ...
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1answer
24 views

Characterization of orthogonal projections in terms of operator norms

I want to show the following equivalence: If $X$ is a Hilbert Space and $P\in B(X)$ (i.e. $P$ is bounded and linear) and $P^2=P$, then $$ (\text{im}\,P)^{\perp} =\ker P\iff ||P||\le 1 $$ I know that ...
4
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1answer
32 views

Spectrum of difference of two projections

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. ...
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2answers
30 views

$A$ is a Hermitian operator on an infinite dimensional Hilbert space and $\langle Ax|x\rangle=0$ for all $x$, prove $A=0$ without the spectral theorem

If $A$ is a Hermitian operator on an infinite dimensional Hilbert space such that $\langle Ax|x\rangle=0$ for all vectors $x$, can we prove $A=0$ without the spectral theorem? The proof seems ...
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10 views

Spectral bands of periodic differential operators

I am reading the book "Multidimensional Periodic Schrödinger Operator" (O. Veliev, 2015) which says on page 11: It is well-known the following statements about the spectral properties of $L_{t}(q)$...