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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Partial isometry and projection

The following is a Theorem of Murphy's C*-algebras and operator theory: Let $H_1, H_2$ be Hilbert spaces and $u\in B(H_1,H_2)$. If $u^*u$ is a projection, then $uu^*u=u$. To show it, for $\xi\in H_1$...
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+50

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F \...
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1answer
16 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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$S$ 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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How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
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+500

Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? Context: ...
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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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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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1answer
70 views

Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick

I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the ...
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1answer
57 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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1answer
35 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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0answers
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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51 views

If $Q$ is an operator on a Hilbert space $U$, $(e_n)$ is an ONB of $U$ consisting of eigenvectors of $Q$, then $(Q^{1/2}e_n)$ is an ONB of $Q^{1/2}U$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ ...
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1answer
36 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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2answers
85 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
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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1answer
36 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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1answer
40 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
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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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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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0answers
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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
197 views

the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
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1answer
34 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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1answer
40 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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3answers
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Nontrivial closed ideal of $\mathbb{B(H)}$, $\mathbb{H}$ is a non-separable Hilbert space.

$\mathbb{H}$ is a non-separable Hilbert space. Give an example of nontrivial closed ideal $I$of $\mathbb{B(H)}$, that is different from $\mathbb{B_0(H)}$ which is the ideal of compact operators. Any ...
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0answers
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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0answers
28 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
20 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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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
461 views

Commutating operators and tensor products

I have this lecture slides about commutators and tensor products, but there is one part that I don't understand: The operators and are commuting operators on the tensor product and their sum has ...
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1answer
27 views

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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Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
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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
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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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Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
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1answer
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The difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...
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2answers
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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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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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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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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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1answer
23 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 ...
3
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1answer
25 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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1answer
28 views

If $A_j$ is an increasing family of Hermitian operators such that $A_j\nearrow A$ weakly, for $A=\mathrm{LUB}A_j$, then $A_j\rightarrow A$ strongly.

I am trying to prove the following proposition from Berberian's 'Notes on Spectral Theory': Proposition 1: If ($A_j$) is an increasingly directed family of Hermitian operators, and if the family ...
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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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0answers
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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 $L_{t}(q)$...
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1answer
50 views

Elementary proof that $a - 1$ is not invertible, for self-adjoint $a$ with $\lVert a \rVert = 1$

Assume $a \in A$ where $A$ is a unital $C^*$-algebra. If $\lVert a \rVert = 1$ and $a^*=a$ we know that $1 \in \sigma(a)$, the spectrum of $a$. This follows from the fact that $\lVert a \rVert = r(a) =...
2
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1answer
53 views

How can we compute the square root of an operator of the form $Cv=\sum_{n\in\mathbb N}\langle v,e_n\rangle_Ve_n$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $V$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ and that the inclusion $\iota$ is Hilbert-Schmidt $C:=\iota^\ast$ denote the ...