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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Norm of quotients

$\newcommand{\Ker}{\operatorname{Ker}}$ Let $X$ be Hilbert space and let $T:X\to X$ be a bounded operator. Define the operator $S: X/\Ker T \to X/\Ker T$ via $S(x+\Ker T)=Sx+\Ker T$. I can show that ...
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104 views

Inversion in a unital C* algebra

Let's say that in a unital C* algebra, we have $b \geq a \geq 0$ and $a$ is invertible. Then $b$ is also invertible. Can we conclude that $a^{-1} \geq b^{-1}$? If so, why? Can any related ...
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127 views

Is the closure of a symmetric operator unique?

Let $T:D(T)\rightarrow H$ be a densely defined symmetric operator in a Hilbert space H. The closure $\overline T$ of $T$ is defined as the operator whose graph $G(T)$ is the closure of the graph of ...
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1answer
314 views

Example of a quasinilpotent operator

Can anybody please give me an example of a quasinilpotent operator $T$, i.e. an operator such that $\sigma(T)=\{0\}$ on $l_2$ such that it has finite dimensional but non-trivial kernel and is not ...
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1answer
255 views

A specific example of the GNS construction

In an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X$. ...
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307 views

Singular-value inequalities

This is my question: Is the following statement true ? Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators. For every $n\in\mathbb{N}$ the following inequality holds: ...
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185 views

Operator norm of the sum of a finite collection of bounded linear operator

I recently got some difficulty with my homework question. The question is: Let $T_1,\dots,T_N$ be a finite collection of bounded linear operators on a hilbert space $H$, each of operator norm $\le ...
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Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
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1answer
276 views

What are the requirements for a “test” function to show operators commute?

To show that two operators $\hat{A}$ and $\hat{B}$ commute, we can check whether $\hat{A}\hat{B}f(x)$ = $\hat{B}\hat{A}f(x)$. My question is regarding the function $f(x)$. To check that $\hat{A}$ and ...
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106 views

If a unital strictly continuous homomorphism $M(A) \to M(B)$ is injective on $A$, is it injective?

Let $A$ and $B$ be C* algebras and let $\varphi : M(A) \to M(B)$ be a unital $*$-homomorphism of their multiplier algebras. Suppose, in addition, that $\varphi$ is strictly continuous (of course, norm ...
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558 views

Hellinger-Toeplitz theorem use principle of uniform boundedness

Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
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263 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
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116 views

A Marcinkiewicz approach

The problem was to prove the following that the operator $$Tf(x)=\int_{\mathbb{R}^N}\frac{f(y)}{|x-y|^\alpha}dy$$ Is continuous from $$L^1 \to \ L_\mathrm{Weak}^{p}$$ where $0<\alpha<N$ and ...
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250 views

Eigenfunctions/Invariance of generic convolution operators

Suppose we are given a convolution operator $$ \mathcal{K}[f\,](t):=\int K(t-s)f(s)ds $$ acting on $f\in H_1$ where $H_1$ is a vector space with orthonormal basis $\{\phi_n(t)\}_{n=0}^{N-1}$. If ...
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1answer
45 views

Vanishing ratio of norms implies vanishing ratio of individual elements?

Consider two vectors $x,y \in \mathbb{R}^n$ be parameterized by a value $t>0$, and suppose that $$\lim_{t \rightarrow 0} \frac{|x(t)|}{|y(t)|}=0,$$ where $|\cdot|$ denotes the standard Euclidean ...
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265 views

Is the Neumann series of a real semidefinite matrix asymptotic?

Let $A \in \mathbb{R}^{m \times m}$ be a real symmetric negative-semidefinite matrix and consider the Neumann series $$\sum_{k=0}^\infty t^kA^k = I + tA + t^2 A^2 + \cdots$$ where $t > 0$ is a ...
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4answers
119 views

Spivak Exercise involving operator norm

The exercise as stated: If $T:\mathbb{R}^{m}\to \mathbb{R}^{n}$ is a linear transformation, show that there is a number $M$ such that $|T(h)|\leq M\cdot |h|$ for all $h\in \mathbb{R}^{n}$. Hint: ...
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1answer
70 views

Extension of an operator defined on (not necessarily closed) subspaces

Is there always an extension of an operator $T:U\rightarrow W$, defined on (not necessarily closed) subspaces of the infinitedimensional Hilbert spaces $H\supseteq U,L\supseteq W$, to the operator ...
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1answer
90 views

Can the range of this operator be closed?

Given an operator $T:H\rightarrow L$, where $H$ is a finite-dimensional Hilbert space and $L$ an infinitedimensional one, is the range of $T$, $T(H)$, a closed set in $L$ ? I know that the image ...
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Question about Angle-Preserving Operators

This an exercise out of Spivak's "Calculus on Manifolds". Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this. Given ...
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172 views

Why Strongly Continuous Representations?

When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi ...
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373 views

Norm of the sum of projection operators

Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an ...
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1answer
46 views

Does this sequence of operators in Hilbert space stop at $\text{rank}T+1$ steps?

Let $\left(T_{n}\right)_{n}$ be a sequence of operator in a infinitdimensional Hilberspace $H$, defined as restrictions of an operator $T:H\rightarrow H$, on smaller and smaller subsets, by the ...
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150 views

Cauchy+pointwise convergence $\Rightarrow$ uniform converges (for an operator in a Hilbert space)

Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$, is Cauchy (with respect to the operator norm) and that there is an operator $L$, such that ...
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1answer
119 views

Has this operator $0$ as an eigenvalue / where is my error?

I know of a theorem that tells me, that every compact linear operator on an infinitedimensional Hilbert space has to have the eigenvalue $0$. On the other hand I have the operator \begin{eqnarray*} ...
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1answer
65 views

How to lift from $B(X)/\mathcal{K}(X)$ to $B(X)$?

I am trying to solve the following problem: $T\in B(X)$. If each $\lambda\in \sigma(T)\backslash\{0\}$ is an isolated point in $\sigma(T)$ and the riesz projection corresponding to $\lambda$ is of ...
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3answers
520 views

A compact operator in $L^2(\mathbb R)$

Let $g \in L^{\infty}(\mathbb R)$. Consider the operator $$ \begin{split} T_g\colon & L^2(\mathbb R)\to L^2(\mathbb R) \\ & f \mapsto gf \end{split} $$ Prove that $T_g$ is compact ...
8
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332 views

Selfadjoint compact operator with finite trace

I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$. Can we conclude that $T$ is trace ...
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Why do we distinguish the continuous spectrum and the residual spectrum?

As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension. If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...
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Similarity orbit of compact operators

I am considering a problem connecting the spectra of compact operators to larger class of operators. Since spectra are invariant under similarity, I wonder whether there is a good reference on ...
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1answer
247 views

Schwarz inequality for unital completely positive maps

I came across the following form of Schwarz inequality for completely positive maps in Arveson's paper: Let $\delta:\mathcal{A}\to\mathcal{B}$ be a unital completely positive linear map between ...
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1answer
195 views

Two questions from Dixmier's book on Von Neumann algebras

It seems something is going wrong with the preview I linked in some of my previous questions, so I will just type out the question. I am having trouble with Dixmier's proof of Corollary 5 on p. 46. ...
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350 views

Uniform boundedness principle statement

Consider the uniform boundedness principle: UBP. Let $E$ and $F$ be two Banach spaces and let $(T_i)_{i \in I}$ be a family (not necessarily countable) of continuous linear operators from $E$ into ...
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Adjoint of the infinitesimal generator of a stochastic process

I need help seeing that $$ \mathcal{L}^* g = -\frac{\partial (bg)}{\partial x} + \frac{1}{2}\frac{\partial^2(\sigma^2g)}{\partial x^2} $$ is the adjoint operator of $$ \mathcal{L} = b\frac{\partial ...
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1answer
139 views

A simple question about the open mapping theorem

$X, Y $ : Banach space, $T : X \to Y$ : linear bounded operator, onto. I'm studying open mapping theorem, but how can I prove this? If $B_Y (0, \epsilon_1 ) \subset \overline{T(B_X (0, \epsilon_2 ...
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1answer
136 views

why isn't “functional operator” a contradiction?

Consider the term "functional operator". My understanding was that: (a) An operator in this context refers to a mapping from one vector space to another vector space. (b) A functional is a mapping ...
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216 views

Why are compact operators 'small'?

I have been hearing different people saying this in different contexts for quite some time but I still don't quite get it. I know that compact operators map bounded sets to totally bounded ones, that ...
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1answer
81 views

What are vector states?

I am reading some papers from the 70s on operator theory. I come across the term 'vector state' of a $C^*$-algebra quite often. It is a little bit confusing. Wikipedia redirects to quantum state ...
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51 views

Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
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When the ultrastrong closure of a *-algebra contains the double commutant

As lemma 6 on p.44 of Dixmier's book on Von Neumann algebras, he states that if $A$ is a *-algebra (i.e. possibly without identity, not necessarily closed in any topology) of operators in $B(H)$ such ...
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1answer
486 views

An example of bounded linear operator

Define $\ell^p = \{ x (= \{ x_n \}_{-\infty}^\infty) \;\; | \;\; \| x \|_{\ell^p} < \infty \} $ with $\| x \|_{\ell^p} = ( \sum_{n=-\infty}^\infty \|x_n \|^p )^{1/p} $ if $ 1 \leqslant p <\infty ...
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Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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What does $(B+I)/I\sim B/(B\cap I)$ tell us?

Let $A$ be a $C^*$-algebra in which $B$ is a $C^*$-subalgebra and $I$ is a closed ideal. In several books on $C^*$-algebras I have encountered the following: $(B+I)/I$ is $*$-isomorphic to ...
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1answer
156 views

norm of operator in Hilbert space and complex conjugate Banach space

Let $E$ and $F$ be complex Banach spaces. We denote by $\overline{E}$ the compex conjugate of $E$, that is, the vector space $E$ with the same norm but with the conjugate multiplication by a complex ...
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Two questions about ultraweak and ultrastrong topology from Dixmier

You could reference Dixmier's book on Von Neumann Algebras p.42 Theorem 1 and its proof to know the entirety of the context. Otherwise, the most relevant things are below: Let $M$ be an ultraweakly ...
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Equivalent Definitions of the Operator Norm

Would you give me a proof of the equivalence of these ones? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ ...
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2answers
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Proof that operator is compact

Prove that the operator $T:\ell^1\rightarrow\ell^1$ which maps $x=(x_1,x_2,\dots)$ to $\left(x_1,\frac{x_2}{2},\frac{x_3}{3},\dots\right)$ is compact. For an arbitrary sequence $x^{(N)}\in\ell^1$ ...
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60 views

Why operator systems contain an abundant of positive elements?

I am reading Sec33 of Conway's A Course in Operator Theory, according to his definition, An operator system is a linear manifold $\mathcal{S}$ in a $C^*$-algebra such that $1\in\mathcal{S}$ and ...
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1answer
263 views

Different types of continuity for operators on Hilbert spaces

In chapter one of K-theory and $C^*$-algebras, a Friendly Approach, the author gives a very brief discussion about several types of continuity of operators between Hilbert spaces. Let $T: ...
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161 views

Duals via a Bilinear map

Let $E$ and $F$ be normed vector spaces. Then if $B$ is a bounded bilinear form on $E \times F$ then every $y \in F$ defines a bounded linear functional $f_y$ where $f_y(x)=B(x, y) \forall x \in E$. ...