Questions tagged [operator-theory]

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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37 views

Homotopy between idempotents of small difference

Let $A$ be a unital $C^*$-algebra. It is known that if $p$ and $q$ are projections in $A$ with $$\|p-q\|<1,$$ then $p$ and $q$ are homotopic through a path of projections. Question: Does a similar ...
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14 views

Show operator norm of a linear transformation satisfies condition

Question: Show that the operator norm of a linear transformation T on $\Bbb R^n$ satisfies $||T|| = \max_{|x|=1} |T(x)|$ = $sup_{X \neq 0} \frac{|T(x)|}{|x|}$ I understand that by definition, $||T|| =...
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22 views

Proof that $\frac{2x}{1+x}\leq f(x) \leq \frac{1+x}{2}$ for standard operator monotone function $f$

I am reading a text on operator monotones, defined as Definition 1 (Operator Monotone) A function $f:I\to\mathbb{R}$ defined on an interval $I \subset \mathbb{R}$ is said to be operator monotone if $$...
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2answers
50 views

Find the eigenvalues of the following operator [closed]

Find the eigenvalues of the following operator. $$\cos\left(ih\dfrac{\mathrm \partial}{\partial\varphi}\right),$$ where $h$ is the Plank constant and $\varphi$ is some variable. I was thinking of ...
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10 views

The sum of of a monotone lipschitz operator and a Lipshitz operator

Is there a result which says that the sum of a monotone lipschitz operator and a Lipshitz operator is a monotone operator?
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Showing an operator to be trace class if a sum converges absolutely

In my functional analysis class, we are studying trace class operators and I have just met this exercise For a Hilbert space $H$ and a positive self-adjoint operator $T$ and an orthonormal basis $\{\...
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30 views

Prove convergence of series under trace-norm topology

Let $H$ be a separable Hilbert space, $T(H)$ the set of trace-class operators, and $D(H)\subset T(H)$ the set of density operators (i.e., positive and having trace 1). For any unit vector $\vert \...
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1answer
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Do natural powers of convergent operators also converge

I was thinking about this just now. If we are in a complex Hilbert space $H$ and we have a sequence of self-adjoint operators converging in norm, $A_n \to A$, is it also true that $A_n^m \to A^m$ for ...
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1answer
37 views

Proving square roots of convergent positive operators are convergent

In my functional analysis class, we are studying positive self-adjoint operators and their square roots. I have the following exercise Let $A_n \to A$ be a sequence of positive self-adjoint operators ...
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19 views

Is the sum of a shift operator and a compact operator is invertible?

Let $T$ be the right shift operator, that is, $T(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)$, suppose $E$ is a compact operator from $\ell^{\infty}\rightarrow \ell^{\infty}$, can $T+E$ be an invertible ...
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1answer
21 views

Spectrum of an element in an ultraproduct

In the proof of $(a) \Rightarrow(b)$, we have $\pi_{\omega}(f(\phi(1))=0$. It contradicts the fact that the spectrum of $\pi_{\omega}((\phi(1))$ is $[0, 1]$. I don't understand this statement. I know ...
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45 views

Must a linear operator that vanishes on the basis vectors vanish everywhere?

Let $H$ be a separable Hilbert space with an orthonormal basis $e_1,e_2,\dots$. Consider a linear functional $L: H\to \mathbb{R}$, which may be bounded or not, that obeys $L(e_i)=0$ for all $e_i$. ...
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22 views

An example of a compact operator which is not trace class

In my functional analysis class, we are studying trace class operators where we use the following definitions For a Hilbert space $H$ and a positive self-adjoint operator $T$ and an orthonormal basis ...
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11 views

Transpose of compact operator is compact

Let $X,Y$ be normed spaces. $T:X\rightarrow Y$ is compact and $T^*:Y^*\rightarrow X^*$ is its transpose. Let $B$ be the unit ball in $Y^*$ and $\epsilon>0$ be given. Since the image of the unit ...
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1answer
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Difficulty with: If $P,Q \in \mathcal{L}(H)$ and $0 \leq P \leq Q \implies ||P|| \leq ||Q||$

In the book "Observation and Control for Operator Semigroups" of Marius Tucsnak and George Weiss page 392. I found the following statement If $P,Q \in \mathcal{L}(H)$ and $0 \leq P \leq Q \...
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1answer
19 views

Image of left and right shift operators

Find images of right and left shift operators respectively as $$R(x)=(0,x_1,x_2,\dots), \quad L(x)=(x_2,x_3,\dots) \quad \text{for } x=(x_1,x_2,\dots).$$ Operators are defined on any sequence space ...
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31 views

Prove that the resolvent set of an operator is an open set

I need to prove that the resolvent set of the operator $T$ , is an open set. I saw the proof in the book of Bernard Beauzamy and I understand the work done there except one part. The proposition ...
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28 views

Operator Norm of the Pullback

Let $U,V$ be normed vector spaces. Let $\phi: U \to V$ be bounded and linear. Let the (continuous) pullback of $\phi$ be $$\phi^\ast : V^\ast \to U^\ast$$ $$\phi^\ast(f) = f \circ \phi$$ which is ...
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Induced spectral norm vs induced Frobenius norm for positive maps

Chapter 2 of Bhatia "Positive Definite Matrices" book defines norm of map $T$ between symmetric positive definite matrices as induced spectral norm, rather than induced Frobenius norm. This ...
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1answer
21 views

How to prove Projection operators are Hermitian/positive?

I'm using Dirac notation here. Given only that a projection operator is defined by the property $P=P^2$, prove that $P$ is a positive operator on the Hilbert Space, i.e. $ \langle v|P|v\rangle \geq 0 \...
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1answer
36 views

$*$-homomorphism induced by a c.p.c order zero map

The $*$-homomorphism $\Phi:C_0((0,1])\otimes A\rightarrow B$ is defined in (3). If $f$ is any nonzero element in $C_0((0,1])$, how to define $\Phi(f\otimes a)$? where $a\in A$
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1answer
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Dimension of $B(H)/K(H)$

Let $H$ be an infinite dimensional separable Hilbert space. I'm looking for an easy way to show that the dimension of the quotientalgebra $B(H)/K(H)$ is infinite. Attempt: If the dimension is finite, ...
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11 views

Quasi-interior points of a sublattice

Let $1\leq p< \infty$ and $E= L^p(\Omega,\mu)$. If $A \subseteq \Omega$ is a measurable set and $T: E \to E$ is defined by $Tf=\mathbf{1}_Af$, then find the quasi-interior points of $\operatorname{...
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1answer
26 views

Kadison's transitivity theorem application in a proof (Murphy's $“C^*$-algebras and operator theory”)

Excuse the long post, I just wanted to make the post self-contained so that those who don't have Murphy's book can also understand my question. Consider the following fragment in Murphy's book (p161): ...
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1answer
26 views

Gaps in derivation of thermodynamic property equations.

If $h=h(T, P)$. Does $ dh = c_pdT + \left[v - T\left(\frac{\partial v}{\partial T}\right)_P \right]dP \Rightarrow h_2 - h_1 = \int_{T_1}^{T_2} c_pdT + \int_{P_1}^{P_2}\left[v - T\left(\frac{\partial v}...
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10 views

Bounded linear extension and closure of domain/codomain

It's known that if $T\colon X\rightarrow Y$ is a bounded linear operator between normed spaces and $Y$ is complete, then $T$ admits a norm-preserving extesion to $\overline{X}$, the closure of $X$. ...
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40 views

Find conditions when derivative of the function is zero [closed]

I am studying the properties of the functions and its derivatives. And I came up with the following questions: Let $M$ be a functions such that $M(x,y)\leq M(x,z)+M(z,y)$. Set F(z)= M(x,z)+M(z,y). ...
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19 views

Equivalent characterization of compact operator

Let $T: X \to Y$ be an operator between two normed vector spaces. Recall that $T$ is said to be a compact operator whenever $T$ maps the closed unit ball $U$ of $X$ onto a norm relatively compact ...
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1answer
16 views

Continuity of the composition of operator-valued continuous functions

Let $M$ be a metric space, $E_i$ be a $\mathbb R$-Banach space and $f_i:M\to\mathfrak L(E_i,E_{i+1}$. If $x\in M$ and $f_i$ is continuous at $x$, can we show that $f_2f_1$ is continuous at $x$? We ...
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54 views

How to prove that every hereditary C*subalgebra of a non-elementary simple C*algebra has infinite dimension?

$A$ is said to be elementary if $A$ is isomorphic to some $\mathcal K(\mathscr H)$. A C*-subalgebra $B$ is said to be hereditary if for every $0\leq a\leq b\in B$ we have $a\in B$. I can see that $A$ ...
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7 views

Calculating infinitesimal generator on two-composite Markov process

I have a problem where I consider a material consisting of layers, which are randomly distributed. $Y(z)$ is assumed to be Markovian. In each $z$ interval the process $Y(z)$ takes values that are ...
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9 views

Norm of a system of and integral operator

Let $T$ be the vector-valued integral operator defined by $$ \begin{eqnarray*} T &:&L^{2}(0,1)\times L^{2}(0,1)\longrightarrow L^{2}(0,1)\times L^{2}(0,1), \\ (u,v)^{t} &\mapsto &T(u,...
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1answer
25 views

Translation operator unitarily equivalent to multiplication by exponential

This is part of a problem from Hall's book "Quantum Theory for Mathematicians". Determine the unitary operator $U:L^2(\mathbb{R^n})\to L^2(\mathbb{R^n})$ (unique up to a constant) such that ...
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36 views

Connes' emdedding conjecture

I wonder whether the Connes' emdedding conjecture has been solved compeltely. According to this link Quantum information theory and nonlocal games, the poster mentioned that the conjecture was false ...
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1answer
46 views

Is every bounded operator part of a $C_0$-semigroup?

Let $X$ be a Banach space and $B \in \mathcal{B}(X)$ be a bounded linear operator on $X$. Is there necessarily a $C_0$-semigroup $T$ such that $B = T(t)$ for some $t$? There might be something obvious ...
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46 views

How to bound marginal changes to Toeplitz projections onto different Hilbert spaces?

Notation: $X$ and $Y$ are vectors in $\ell^2$. Let $P_X$ denote the projection operator onto the vector space spanned by $\{L^jX\}_{j=0}^{\infty}$ where $L$ is the right-shift operator. $P_X$ is a ...
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1answer
59 views

Spectrum of a positive operator in $B(H)$.

We know that for $T\in B(H)$. If $T$ is positive, then $T$ is self-adjoint and $\sigma(T)\subset R^{+}$. Do we have the inverse ie: if $T$ is self-adjoint and $\sigma(T)\subset R^{+}$. $T$ is positive....
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7 views

Existence of LOCC orthogonal projection measurement.

Let $M = \{M_x\}_{x\in X}$ be a POVM on $\text{End}(H_{AB})$. By Naimark's dilation theorem there exists an orthogonal projection measurement $P = \{P_x\}_{x\in X}$ on some larger space yielding the ...
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1answer
89 views

Question about polar decomposition. Is the book wrong?

An introduction to the classification of amenable C*-algebra. page 140 Lemma 3.5.1 Let $x\in A$ with the polar decomposition $x=u|x|$ in $A''$ and $B=\overline{x^*Ax}$. Then $ub\in B$ for every $b\in ...
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1answer
31 views

Square of an approximate unit is an approximate unit

Let $A$ be an abelian $C^*$-algebra with approximate unit $(u_\lambda)$. I'm trying to show that $(u_\lambda^2)$ is an approximate unit as well. In particular, I'm stuck at showing that $$\lim_\lambda ...
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2answers
53 views

If $\varphi: A \to B(H)$ is a representation, then $\Vert \varphi(a) \Vert = \Vert \varphi_K(a) \Vert$ (subrepresentation)

Consider the following fragment from Murphy's book "$C^*$-algebras and operator theory": Why is the marked line true? I want to know why $\Vert \varphi(a) \Vert \leq \Vert\varphi_K(a)\Vert$....
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22 views

Linear extention of unitary operator

I have a Real Hilbert space $H$ and an operator $\beta$ over $H$ satisfyng $\beta^{\ast}=-\beta$ and $\beta^2=-1$ (this easily implies $\beta$ would be unitary). Now I consider the complixification ...
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2answers
26 views

Strong closure of an abelian $*$-algebra is abelian.

Let $A \subseteq B(H)$ be a $*$-subalgebra of the bounded operators on the Hilbert space $H$. Let $B$ be the SOT-closure of $A$. Is $B$ again abelian? Let $u,v \in B$. We can find nets $(u_\lambda), (...
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1answer
40 views

Show strong topology on the closed unit ball is metrizable

Let $S$ be the closed unit ball of $B(H)$, the bounded operators on the Hilbert space $H$. I want to show that the relative strong topology on $S$ is metrizable. Attempt: I have already established ...
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29 views

Proving that an operator that commutes with any other operator defined on a euclidean space is an homothecy

I need to show that if $A$ is an operator defined on a euclidean space so that it commutes with every other operator $B$ defined on the same euclidean space, then $A=\lambda I$. Attempt at a solution ...
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12 views

singular values of operator

Suppose that $T_n$ are operators on an infinite dimensional separable Hilbert space $H$ such that rank of $T_n$ is less than $r_n<\infty$ $\| T_n\|\leq c_n$, where $c_n\to 0$. Let $T=\bigoplus_{n\...
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1answer
37 views

Eigenvectors associated to distinct eigenvalues are orthogonal (with T a normal operator)

Let be V a inner product space over $\mathbb{C}$, $T$ a normal operator in $V$ and $u,v \in V$ two eigenvectors of T corresponding to different eigenvalues. Prove that $u$ and $v$ are ortogonal. I was ...
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36 views

Defining operator norm as the spectral norm of its matrix

Suppose $T$ maps between symmetric positive definite $d$-by-$d$ matrices $X,Y$ as follows: $$\operatorname{vec}(Y)=M\operatorname{vec}(X)$$ $M$ is a symmetric positive definite $d^2$-by-$d^2$ matrix. ...
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21 views

Lim sup nets proof, is it valid?

I just get really nervous around lim sup and nets for some reason... Please help! Problem I was given: Suppose $(x_i)_{i\in I} \subset B(\mathcal{H})$ converges to $x \in B(\mathcal{H})$ in the SOT (...
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32 views

Is this positive map also completely positive?

Can $T$ defined below be shown to be a completely positive map? $$T(A)=E[(I-XX')A(I-XX')]$$ $T$ maps between covariance matrices, X is a (column)-vector sampled from some $d$-dimensional distribution (...

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