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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Show that operator norm of linear map is equal to $a^2$

Let $a>0$. Show that the linear map $A$ that assigns every $x \in C([0,a])$ the function $s \mapsto s \int_0^a x(t) dt$ from $C[0,a]$ has operator norm $a^2$. I'm not really sure where to start ...
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Multiplication operator is densely defined.

I saw the question Multiplication Operator on $L^2$ is densely defined Disintegration by parts answers using the following argument. If $f \perp \mathcal{D} (L)$ then $\frac{1}{m^2+1} f \in \mathcal{D}...
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Estimation of operator norm of the difference between identity and Gram matrix?

I have a "tall" matrix $W\in\mathbb{R}^{N\times n}$, $N>n$, with its operator norm being upper-bounded by some constant $C>0$, i.e., $\|W\|_{2\rightarrow 2}\leq C$. What I am trying to ...
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Essential spectrum of a projection

For any $T\in B(H)$, the essential spectrum $\sigma_e(T)$ of $T$ is the subset of the spectrum $ \sigma(T)$ of $T$. If $P$ is a projection, we have $\sigma_e(P)\subset \sigma(P)=\{0,1\}$. My question ...
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Prove for $\|\left[\begin{array}{cc} A&0\\ 0&B \end{array} \right]\|\leq \max\{\|A\|,\|B\|\}$

For this lemma Lemma Let $A$ and $B$ be positive operators on $H$. If $T$ is the operator matrix on $ H ⊕ H$ defined by $T =\left[\begin{align} A&C^*\\ C&B \end{align} \right]$ Then $$\|T\|\...
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Alternative reference request for Kato's Perturbation Theory

I am studying Analytic Perturbation Theory from Kato's book.But sometimes I find it really difficult to follow.(e.g. In Chapter 2 , he directly talks about algebraic singularities without defining it ...
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Norm inequalities for sums of two basic elementary operators

I'm working on this paper. What I'm intersted in is this theorem: where $M_{A,B}(X)=AXB$ I don't know why we can find those two sequences $X_n$ and $x_n$, either I'm finding difficulties to show the ...
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Question regarding conjugate operators and the harmonic operator.

Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$ I'...
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dilatation analytic

I have seen that the spectrum of the operator $T=-\frac{d^2}{dx^2}+x^2$ is the $\{2n+1,n \in \Bbb{N}\}$ and by dilatation $x=r y$ the spectrum of the operator $T_r=-r^{-2}\frac{d^2}{dx^2}+r^2x^2$ is ...
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Confusion about Rudin Submodule

Here $V_1, V_2$ are multiplication operator by coordinate functions. The above Theorem is a result obtained by V. Madrekar in 1988. In the article "A brief survey of operator theory on $H^2(\...
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Defining free Dirac operator on $\big(L^2(\mathbb{R}^n)\big)^{n+1}$

So I have been considering free Dirac operator $D_0:(L^2(\mathbb{R}^3))^4 \supset (W^{2,1}(\mathbb{R}^3))^4 \to (L^2(\mathbb{R}^3))^4$ given by formula \begin{equation} D_0 = \left[\begin{matrix} ...
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pure states on B(H) space

Let $A$ be a $C^*$-algebra, we define a state $U$ on $A$ as a positive linear functional with this property: $U(1)=\|U\|=1$. my problem is to characterize pure states on $B(H)$ space. show that for ...
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Spectrum analysis

I have a problem: Given the Hamiltonian $ H_n= -\frac{d|}{d^2}+V_n $ where $ V_n= a n e^{-n|x|}; a\in\Bbb R, n=1,2,....$ 1)Domain in which the operators are self adjoint; 2)study and list the ...
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Commutant of corner of $*-$algebra on a hilbert space is corner of commutant

Let $A$ be a $*-$algebra on a Hilbert space $H$ and $p$ be a projection in $A'$, where $A'$ is the commutant of $A$, that is, $$A':=\{u \in B(H): ua=au~\text{ for all }~a \in A\}.$$ If also $p \in A''$...
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what is compactness argument

I'm working on this paper and I don't know what is meant by compactness argument in the proof of corrollary 4 page 226 which said that: the function $\lambda \to \|B-\lambda A\|$ (where A and B are ...
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Maximal abelian von Neumann algebra and cyclic vectors

I know the fact that if $A\subset B(H)$ is a maximal abelian von Neumann algebra and $H$ is separable, then $A$ will have a cyclic vector. This result is proved in Conway's book "A course in ...
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Set of densities of spectral measures

Let $(X,\mathbb{A},E)$ be a spectral measure on Hilbert space $H$ and $f,g \in H$. We can define scalar measures $\mu_{f,g}(\delta)=(E(\delta)f,g)$ and $\mu_{g}(\delta)=(E(\delta)g,g)$. Now fix $g$. ...
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Asymptotics of singular values ​of the operator

In space $L^2(B_R(0))$ consider operator $Tx(t)=\displaystyle\phi(t)\int\limits_{B_R(0)}\cos(|t|^{1/2}|s|^{1/6})\psi(s)x(s)ds$, where $\phi, \psi \in L^2$. It is clear that this operator is compact. ...
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$\bigoplus_\lambda A_\lambda$ is strongly closed in $B\left(\bigoplus_\lambda H_\lambda\right)$.

Let $(H_{\lambda})_{\lambda\in \Lambda}$ is a family of Hilbert spaces and $A_{\lambda}$ is a von Neumann algebra on on $H_\lambda$ for each index $\lambda$. Then prove that the direct sum $\bigoplus_\...
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Is the orthogonal complement projector a projection operator?

Let $X$ Hilbert space and $U \subset X$ convex closed. Then we can define the projection operator on $U$ $P \colon X \to X$ such that $P x \in U$ is the orthogonal projection of $x \in X$. Moreover $P$...
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conditional expectation onto discrete MASA

In the above screnshot, $X$ is a MASA in $L(H)$ and $Z\in L(H)$. I tried to prove (3.1) and met with a question. When we take $Z =\begin{pmatrix}A & B\\C & D\end{pmatrix} $, where $A\in L(H_d),...
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How is the restriction of the disk algebra to the boundary seen as a subalgebra?

The disk algebra is the set of continuous functions $f: D \to \mathbb C$ where $D$ is the closed unit disc in $\mathbb C$ and $f$ is analytic on the interior of $D$. It is endowed with the $\sup$-norm....
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If $T$ is a compact operator and $S$ is a bounded operator then $TS-ST$ cannot be the identity operator.

Let $T:L_2[0,1] \to L_2[0,1]$ be a compact linear operator and $S:L_2[0,1] \to L_2[0,1]$ be a bounded linear operator. Prove $$TS-ST \neq e$$ where $e$ is the identity operator. My solution. I am ...
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Bounded inverse for a closed range closed operator

I am dealing with an unbounded operator $T$ on an Hilbert space $H$. I am interested in proving that it has a bounded inverse $T^{-1}$. I managed to prove that the operator is closed and self-adjoint. ...
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Banach spaces and surjective operators

Let $X,Y$ be two Banach spaces and $T \in L(X,Y)$ be surjective. Then there exists a constant $C>0$ so that for every $y\in Y$ there exists a $x\in X$ with $Tx=y$ and that the following is true: $||...
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Confused on two parts of Proof in C* algebras by Murphy

I'm working through the proof of the following theorem from C* algebra by Murphy. For context, $T_{\varphi}: H^2(T) \longrightarrow H^2(T)$ is given by $T_{\varphi}(f) = p(\varphi f)$ for $p: L^2(T) \...
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1 answer
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Quotient space and continuous operator

Let $X,Y$ be normed spaces and $T \in L(X,Y)$. We define $K=\ker(T)=\{x \in X:Tx=0\}$. Show that for $U:X/K \rightarrow Y, x+K \rightarrow Tx$: $U\in L(X/K,Y)$. Well, by definition a linear operator ...
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Understanding the proof of Fuglede's theorem

I am trying to understand this proof of Fuglede's theorem on wikipedia : Everything was making sense until the last sentence: "Considering the first-order terms in the expansion for small λ, we ...
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How can we expand $(f(x)*d/dx)^n$? And why? [closed]

Which one is the correct expansion of $(f(x)d/dx)^2$? $(f(x))^2 (d^2/dx^2)$ or $(f(x)d/dx)(f(x)d/dx)$? And why?
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Definiteness of Nemytskii operator on space of continuous functions

Let consider a function $f\colon \overline{\Omega} \times \mathbb{R} \mapsto \mathbb{R}$, where $\overline{\Omega}$ is subset of $\mathbb{R}^n$. $\overline{\Omega}$ can be either bounded or ...
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Prove that $\frac{1}{2}N_1(A_1,A_2)\leq N_2(A_1,A_2) \leq N_1(A_1,A_2)$ where $N_1$ and $N_2$ are two norms

Let $A_1$ and $A_2$ two bounded linear operators on a complex Hilbert space $E$. I want to prove that $$\frac{1}{2}N_1(A_1,A_2)\leq N_2(A_1,A_2) \leq N_1(A_1,A_2),$$ where $$N_1(A_1,A_2)=\...
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Determine whether a self-adjoint operator is negative definite or not

Let $Q$ be an idempotent in $L(H)$. Suppose there exists two sequences of unitaries $\{U_n\}$ and $\{V_n\}$ such that $X$ is the norm limit of $U_nQU_n^*+V_nQV_n^*$, and suppose that $Q+Q^*\cong 2P\...
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1 vote
1 answer
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Given $[A,B]=0$ then $[A,f(B)]=0$

Given $A$ and $B$ two operators that commute ($[A,B]=0$) then $A$ commutes with an arbitrary function of $B$ I recently saw this property of the commutators on a quantum mechanics course. We didn't ...
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Spectral resolution of the identity of $S = U^*TU$

Suppose $T,U$ are bounded operators on a Hilbert space $\mathcal{H}$ with $T$ normal and $U$ unitary. Now we define $S = U^*TU$. Then I want to find the spectral resolution of the identity $E_S$ for $...
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Solution of the operator equation $AX + X^*A=0$

I'm searching for the solution of the following equation $$ A X + X^* A =0 $$ where $A$ is hermitian and invertible bounded operator, while $X$ is bounded as well. I was using the following results (...
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decomposition of an idempotent in $L(H)$

The above screenshot is from J.C.Bourin and E.Y Lee's paper. When reading the proof of the above Proposition, I met with a problem: How to decompose the Hilbert space $H$ as the direct sum of $H_s$ ...
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2 votes
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The enveloping $C^*$ algebra of $C^k[0,1]$.

We know that $C^k[0,1]$ is an abelian $*$-Banach algebra, but it is not a $C^*$ algebra in general unless $k=0$. I wonder what's the enveloping $C^*$ algebra of $C^k[0,1]$? I guess it might be $C[0,1]$...
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Finding inverse of the operator

I am very new to finding inverses of the operators in the functional analysis. I have an exercise question in my university course, and I am trying a lot to get a solution for it. It is a complicated ...
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Proof of minimum value not equal to zero in linear bounded operator mapping(Hilbert space)

I am new to operators and Hilbert spaces in functional analysis. I am trying very hard to prove the below question but could not do it. It is an exercise problem in my university course. Could anyone ...
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2 votes
1 answer
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Equivalence between two definitions of hermitian adjoint

Given the two definitions of hermitian adjoint: $(1): <\Psi|A|\Phi>=(<\Psi|A^+|\Phi>)^*$ $(2): <\Psi|A\Phi>=<A^+\Psi|\Phi>$ I want to show that they are equivalent However I ...
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2 votes
1 answer
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Question about Hille Yosida Theorem proof

I'm working on Hille-Yosida theorem on Vrabie's book. Here is the statement: In order to prove the sufficiency two lemmas are needed: and Here comes my question, is highlighted in yellow: Why can ...
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How to prove boundedness of a shift operator of an infinite sequence in two directions

Consider X = $l^{p}$ (Z; C) = { (. . . , $x_{−1}$, $x_{0}$, $x_{1}$, . . .) | $x$$_{k}$ ∈ C} I want to show that the right shift operator $Sr$ | ($Sr\hspace{1 mm}x)_{k}$ = $x_{k−1}$, $x$ = $(x_{k})_{...
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1 answer
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Confused on a set inclusion in C* algebras by Murphy

Im stuck on the following part of this theorem: The closed vector subspaces of $L^2(T)$ invariant for the bilateral shift $v=M_{z}$ (for $z: T \longrightarrow \mathbb{C}$ the inclusion map) are ...
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operator between Banach spaces and its boundeness

I am trying to understand the proof of the following: Let $T:X\to Y$ be an operator between two Banach spaces $X$ and $Y$ and $T$ is linear. Let $\mathrm{dim}\,X=n\,<\infty$. Then $T$ is bounded. ...
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1 vote
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Von Neumann algebras are generated by its projections

In a lecture about operator theory we used the claim, that the set of projections in a von Neumann algebra $\mathcal M$ is dense in $\mathcal M$, with respect to the operator norm. Sadly that claim ...
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2 votes
1 answer
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an inequality for completely positive maps

Let $A$ and $B$ be $C$*-algebras and $\phi:A\to B$ a completely positive contractive map. I want to show that, for any $a,b\in A$; $$\Vert \phi(ab)-\phi(a)\phi(b)\Vert\leq\Vert\phi(aa^*)-\phi(a)\phi(a^...
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1 vote
1 answer
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Showing 1 cannot be an eigenvalue after rotating an operator

I have a compact self-adjoint positive integral operator $Q:L^2(0, \infty) \to L^2(0, \infty)$ with operator norm $\| Q\| =1$. By the assumptions, we know $1$ is an eigenvalue of $Q$. Let $y\ne 0$ and ...
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1 answer
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How much can the (reduced) group $C^*$ algebra maintain the subgroup structure?

I have read a theorem which says that if $H$ and $G$ are both discrete groups, $G\leq H$, then $C^*(G)$ is a subalgebra of $C^*(H)$ and $C^*_r(G)$ is a subalgebra of $C^*_r(H)$. For now, what I can ...
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1 answer
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Show that operator is bounded and converges pointwisely but not in the operator norm

For every $n \in \mathbb{N}$ let $T_n:c_0 \rightarrow \mathbb{R}$ be defined through $T_n(x_k)_k=x_n$. First, I need to show that $T_n$ is bounded, which is the case if and only if $T_n$ is continuous....
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1 vote
1 answer
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Strong limit from the unitary orbit of $A$

Let $A,X\in L(H)$ with $\Bbb D\subset W_e(A)$ ,where $W_e(A)$ is the essential numerical range of $A$,and $\|X\|\leq 1$. Then there exists a sequence of unitaries $(U_n)_n$ in $L(H)$ such that wot $...
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