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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Uniqueness of Unitary operator

i saw the post "Polar decomposition normal operator" (Polar decomposition normal operator). There was that such a $U$ is unique iff the image of $T$ is dense. Some lines later by the comments there is ...
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Question about domains of unbounded operators

This is a part of a theorem in Rudin's Functional Analysis, in the chapter on unbounded operators. Let $\mathcal M$ be a $\sigma$-algebra in a set $\Omega$, $H$, a Hilbert space and $E:\mathcal ...
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determinant identity for invertible finite rank operators

I am currently reading a paper where the following identity, valid for an invertible finite - rank operator $T \colon \mathscr{H} \to \mathscr{H}$ on a separable Hilbert space, is given: $$ \log \det ...
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Three basic questions about linear operator in a Hilbert space

Just come across three questions in reading a paper. Suppose we are dealing with a Hilbert space of $L_{2}[0,1]$ and all the functions mentioned below are in $L_{2}[0,1]$. Define the operator $A$ by ...
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Finding an isometry between two subspaces of a Hilbert space

So, I'm given a Hilbert space which is the direct sum $H=H_1\oplus H_2$ of two separable Hilbert spaces $H_j$. There is a closed subspace $D\subseteq H$ which satisfies that it is not a subspace of ...
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Interpretation of Fredholm Alternative with respect to PDEs

I have studied the Fredholm Alternative, which states the following: Theorem: Let $K:H \rightarrow H$ be a compact linear operator and let $I$ be the identity operator on $H$. Then: 1.$N(I-K)$ is ...
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Minkowski functional and strange theorem

I have a theorem that says the following: Let X be a normed space and $U\subset X$ a convx subset with $0 \in \text{int(U)}$, then we have: $U$ is absorbing and if $\{x;||x|| < \epsilon\} \subset ...
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Fredholm alternative and orthonormal basis

The following question relates to the Fredholm alternative: Let $K:H \rightarrow H$ be a compact linear operator and let $I$ be the identity operator. Notation: $N$ is the nullspace and $R$ is the ...
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Norm of a matrix equals greatest eigenvalue

How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an ...
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43 views

Pulling Operator Inside Integral

Say $Y$ is a Banach space and you have a family of continuous/bounded operators $L_{x}: Y \rightarrow Y$ for $x\in \mathbb{R}$ and say you have an bounded, smooth map $f(x):\mathbb{R}\rightarrow Y$. ...
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Predual of $\mathcal{B}(K, H)$

Is there a predual of $\mathcal{B}(K, H)$? So, what does the space $X$ look like, such that $X^*=\mathcal{B}(K, H)$.
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If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)

Let $V,W$ be normed vector spaces, and $L(V,W)$ be the space of bounded linear operators. Usually I would only see the statement "If $W$ is Banach, then $L(V,W)$ is Banach.". But Wikipedia writes that ...
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questions about norm of integral operator

The following is a question I came up with when I was studying the same problem in dimension 1 (for which also I have the questions that follows) but I put in generality. Let $U_1, U_2 \subset ...
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181 views

Integral kernel of the resolvent operator

Suppose we have an explicit formula for the integral kernel $k(x,y)$ of an operator $D$ acting on smooth $\mathbb{C}^n$-valued functions defined on an interval $[0,\beta]$, that is $$ Df(x) = ...
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109 views

Why do $S^{1/2}$ and $T^{1/2}$ commute

This question is actualy related to my old question Product and sum of positive operators is positive If $S,T \in B(H)$ are bounded, linear and normal operators on a Hilberspace $H$, i.e. $SS^*=S^*S$ ...
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45 views

Prove this sequence in $L(\ell_1)$ does not converge to zero?

I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow ...
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difference between idempotent and projection operators

in book of conway, functional analysis, section operators on Hilbert space(projection and idempotent) say that a projection is an idempotent such P that $(kerP)=(rangP)^\perp$. but from the next ...
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Upper bound for norm of Hilbert space operator

It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $||T||=\sup_{||x||=1}|\langle Tx,x\rangle|:=M$. It seems that for any bounded operator on ...
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$\gamma-$radonifying operators.

I am reading about $\gamma$-radonifying operators, and came across 2 similar definitions. And I hoped to understand why they are equivalent. Let $H$ be a seperable real Hilbertspace, $E$ banach ...
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67 views

Positive unbounded operator with zero not as an eigenalue

I am currently doing Quantum Mechanics and I am supposed to show that zero is an eigenvalue of a positive operator. I have no knowledge of Functional Analysis at that kind of level, so I was wondering ...
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84 views

completely continuous implies compact

I'm searching for a proof of the fact that if: $T$ is a bounded operator in a reflexive Banach space that maps weakly convergent sequences onto convergent sequences then $T$ is compact. If we let ...
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About the trace class operators and their motivation

What is the motivation for trace class operators? Can anybody suggest the most general and standard reference that includes Schatten $p$ class operators as well? I have the following references. ...
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spectral mapping type norm identity for self adjoint operator

I am currently trying to understand the spectral theorem as given in "Functional Analysis" (Vol.1) by Reed and Simon. Leading to its proof is a preliminary Lemma where I got stuck. It says Let $P(x) ...
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Compact operator and limit

I was wondering about something related to compact operators. If we have a compact operator $T:X \mapsto Y$ and a bounded sequence $(x_n)n$, then we know that there is a convergent subsequence ...
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Are these operators and the fourier transform compact?

I do not want a proof but rather an explanation. I just read that $T_k:L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ such that $(T_kf)(s) = \int_{\mathbb{R}} k(s,t)f(t) dt $ is compact. (in this ...
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operator differential equation

let be the differential equation for the operator 'X' $$ \frac{dX(t)}{dt} = A(t) X(t) $$ the formal solution is the exponential operator $$ X(t)=X(0)e^{ \int_{0}^{t}A(u)} $$ of course i should ...
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Compute the norm of matrix

Let $M$ be $n\times n$ matrix, consisting entirely of 1's. Show, that $\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$.
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MASAs of C* algebras

While studying the $C^*$-algebraic formulation of the recently solved Kadison-Singer problem, I was wondering about maximal abelian subalgebras: Let $\mathcal{A}$ be a unital C* algebra. There seems ...
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100 views

Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
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54 views

Relation of norms of matrices

Let $A$ be $m \times n$ matrix. Let $B=\frac 1n AA^*$, where $A^*$ is a transposed matrix. Let $X_i, I\leq m$ be row-vectors of $A$. Show $$ \|B\|=\frac 1n \|A\|^2\geq \max_{i\leq m}|X_i|, $$ Where, ...
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27 views

A sequence Which is not weakly convergence

Let H be a infinite Hilbert space and $\{e_n\}$ be sn orthogonal sequence of projections in B(H)> Show that $\{ne_n: n\in N\}$ does not admit a subsequence converging to zero weakly. I tried to proof ...
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Measurability of the dilatation operator

I need some help with this question: We consider the dilatation operator: $T: \mathbb{R^{+}}\to \mathcal{L}(L^p(\mathbb{R}),L^p(\mathbb{R}))$ $\;\;\;\;\;\;\delta\to ...
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Inverse of positive operators

Does anyone know how to show this? Let $H$ be a Hilbert space and $A$, $B$ bounded positive operators defined on $H$ such that $A^{-1}: H \rightarrow H$ exists and hence bounded and $A \leq B$. ...
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Properties of the operator $T: f\to f*g$

Let g be the characteristic function of [-1/2,1/2]. $T: f\to f*g$ (convolution). I have managed to prove that T is a linear,bounded,self adjoint,injective operator and it's immage is inclused in ...
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Compact operator in Hilbert spaces $T^2$

I have the following problem: Let H be a Hilbert space a) Prove that if $T: H\to H$ is compact then $T^2$ is compact operator b) Find $S: H\to H$ compact such that $S=T^2$ with T non compact c)If ...
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35 views

A property of the canonical inclusion $i: L^2(0,1)\to L^1(0,1)$

Prove that the image of the canonical inclusion $i: L^2(0,1)\to L^1(0,1)$ is a countable union of closed sets with empty internal part. Can anyone give me any idea on the solution? Thank you in ...
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A simple question about the dimension of subspace.

I have a simple question: Let $A$, $B$ be closed subspaces of banach space $X$ and $B\subseteq A$, if $\dim A/B<\infty$ and $\dim B<\infty$, then $\dim A<\infty$? Why?
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How can I show that given a norm one linear functional on $c_0$ that there is a unique extension to a norm one functional on $\ell_\infty$?

We are given that our Banach space is $c_0 \subset \ell_\infty(\mathbb{N})$ and there is a functional $y^* \in c_0^*$ such that $||y^*|| = 1$. We are guaranteed that this extends, via Hahn-Banach to a ...
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Eigenvalues of the operator $(Tu)(x)=\int_0^x (\int_t^1 u(s)ds)dt.$

Consider the linear operator $T$ in $L^2(0,1)$ defined by: $$(Tu)(x)=\int_0^x \left(\int_t^1 u(s)ds\right)dt.$$ I have managed to prove that it's continuous,self adjoint,compact but now I have to ...
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Does a symmetric operator on a Hilbert space have a symmetric adjoint?

Suppose we have a linear operator $T$, densely-defined on some Hilbert space. If $T$ is symmetric (i.e., $T^*$ extends $T$: notationally, $T\subseteq T^*$) does it follow that $T^*$ is also symmetric ...
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Boundedness of functional

In the setting of $2\pi$-periodic $C^1$ functions (whose Fourier series converge to themselves), and given a linear functional $D:C^1_{\text{per}}\to\mathbb R$ satisfying ...
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In a separable Hilbert space, can you write an operator from $\mathcal H$ to $\mathcal H$ as a column-finite matrix?

In this question, we are representing an operator $T$ as a matrix with respect to an orthonormal basis $\left\{e_n : n \in \mathbb{N}\right\}$. To do so, we let $t_{ij} = \langle T(e_j),e_i\rangle$. ...
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Existence of bounded linear operator with kernel reduced to $\{0\}$

If $X$ and $Y$ are normed spaces, why there must exist a bounded linear operator $T$ from $X$ to $Y$ such that $T(x)$ is not equal to $0$ for all non-zero $x$?
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Compactness of integral operator

I need some help with this exercise. Let $f\in C^0_b(R^2)$ and consider the operator $[T(v)](x)=\int_0^x f(x,y)v(y)dy$ for every $x\in R$. Is this a compact operator $T:C^0[0,1]\rightarrow C^1[0,1]$? ...
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60 views

Please help to understand text about closed operators and extensions

I need help understanding a section of a book I'm reading (Mathematical Foundations Of Quantum Mechanics, by J. von Neumann, Princeton U. Press, 1955, pages 152-153). I have a few questions on two ...
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145 views

An example of a non-closable operator

I've encountered the following: Consider the usual Hilbert space $L^2([0,1],dx)$ and the dense subspace $\mathcal{D}=\mathcal{C}[0,1]$. Define $T$ on $\mathcal{D}$ by $T(f)=f(0)$. This is a ...
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326 views

Orthogonal Projections in Hilbert space

I am stuck with the following exercise about projections in Rudin 12.26. Let $H$ be a Hilbert space $P,Q\in B(H)$ self-adjoint projections (A projection has the property that $P^2=P$), then the ...
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approximate identity of a *-algebra.

I know that a *-algebra does not always have an approximate unit. Why does a non-degnarate *-algebra which is $\sigma-strongly^*$ closed have an approximate unit?
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Von Neumann algebras

Let $\{M_i\}_{i\in I}$ be a family of von Neumann algebras. Let H denote the direct sum$\Sigma_{i\in I} H_i$ of Hilbert spaces $\{H_i\}_{i\in I}$. Every vector $h=\{h_i\}$ in H is denoted by ...
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Problem 21 - Trotter theorem , Reed and Simon

This problem if from Methods of modern mathematical physics I :Functional Analysis, by Reed and Simon: Problem 21: Let $\{A_n\}$ be a sequence of selfadjoint operators on a Hilbert space $H$, and let ...