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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Proving that an operator $K$ is bounded and $||K|| = \max_{0\leq x\leq 1}\bigg\{\int_0^1|k(x,y)|dy\bigg\}$

Define $K:C([0,1])\rightarrow C([0,1])$ by $$Kf(x) = \int_0^1 k(x,y)f(y)dy,$$ where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is continuous. Prove that $K$ is bounded and $$||K|| = \max_{0\leq ...
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52 views

Does this show the norm of this operator is zero?

We have $$T: C[-1,1]:\to \mathbb{R}$$ $$T(f)=\int_{-1}^1 x f(x) dx$$ The norm considered in $C[-1,1]$ is $$||f||=\max_{x\in[-1,1]} |f(x)|$$ So using $$||T||=\inf\{M:||Tf||\leq M||f||\}$$ in this ...
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59 views

Identity plus finite rank has index $0$

I'm supposed to prove the strong Fredholm alternative in the form $$\text{Ind}(1-K)=0$$ for any compact operator $K:H\to H$ where $H$ is a Hilbert space and $$\text{Ind}(T):=\text{dim Ker }T+\text{dim ...
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1answer
27 views

Index of orthogonal projections

I'm reading a paper 'The index of a Pair of Projections' by Avron, Seiler and Simon at the moment and have a question about a definition: What do they mean by $C$ viewed as a map from $\text{Ran ...
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73 views

Multiplication operator with a function non-vanishing on the cantor set

Let $M_f$ be the multiplication operator, which acts on bounded functions $g$ on the unit interval as $g\mapsto fg$, with $f:[0,1]\rightarrow \mathbb{C}$ such that $f$ is nonzero only on the Cantor ...
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106 views

Limit of nth power of operator norm

I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies ...
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1answer
104 views

Calculate the norm of this operator

$C[0,1]=\{ f : [0,1]\to [0,1], f$ continuous$\}$ $||f||_\infty=\max_{t\in [0,1]} |f(t)|$ $T:C[0,1]\to C[0,1]$ defined by $$(Tf)(t)=\int_0^1e^{s+t}f(s)ds$$ Find $||T||$ The usual way to do this ...
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26 views

Piecewise Continuous Fredholm Kernel

Suppose I have a ``Fredholm equation of the second kind" with kernel $$ K(x, y) = 1[x \geq u]k(x, y) $$ where k(x, y) is continuous. Let $f_n(x)$ be a smooth continuous approximation of $1[x \geq ...
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92 views

Show that the operator is bounded in $L_p$

Consider the operator $C$, acting on functions $f$ on the unit circle $S^1 = \left\{ z \in \mathbb C \mid |z| = 1 \right\}$ by the rule $$ (Cf)(z) = \frac{1}{2\pi i} ...
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64 views

equality of two operators…

Please help me with the following problem( give some hints or references): Let $X$ be a Banach space and $B(X)$ be the algebra of bounded linear operators on $X$. Suppose that $A$ and $B$ are two ...
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79 views

Exchangability of inner product and integral in bochner spaces

For the linear operator $e \in \mathcal{L}(V,V^{*})$, and sufficiently small $\delta s \in V := L^2(0,T,L^2(D))$ and $p \in V$ we have; $$ E[\langle e^{*}p, \delta s \rangle_{V}] = \langle E[e^{*} p ] ...
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60 views

Composing Translations and Reflections

Since $f(-x)$ is a reflection of $f(x)$ in the $y$-axis and $f(x+a)$ is a shift of $f(x)$ by $-a$ units, so for the longest time I've assumed $f(-(x+a))$ is a shift of $-a$ and then reflected in the ...
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68 views

rank of $A \otimes B$

For two matrices $A$ and $B$, what would be the rank of $A\otimes B$ as a matrix? Seems to me that $rank(A\otimes B) = rank(A)\cdot rank(B)$. But I don't see an elegant proof...
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71 views

Why is the set of compact operators closed in the space of all bounded operators between Banach spaces?

Let $X$ and $Y$ be Banach space. $B(X,Y)$ is the vector space of all bounded linear maps from $X$ to $Y$. Also, $K(X,Y)$ is the set of all compact operators from $X$ to $Y$. Why is $K(X ,Y)$ ...
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146 views

Holomorphic functional Calculus in Dunford and Schwartz

I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the ...
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31 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
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37 views

Is each element of factor(von Neumann algebra) a linear combination of projection?

Let $A$ be a factor von Neumann algebra. Then every element of $A$ can be writeen as a finite linear combination of projections in $A$. Is it right? I know a little about von Neumann algebra, thanks a ...
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42 views

Matrix-valued functions with lacunary Fourier series

This question is motivated by investigation of the operator space structure of Hankel matrices (which is surely well-known to the experts). Consider a lacunary Hankel matrix, i.e. a matrix ...
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96 views

Norm space, linear operator exercise, help please!

$f \in L_2[a,b]$ $Uf(s):=\int_a^bk(s,t)f(t)dt$ $k(s,t):[a,b]^2\to R $ continuous. show 1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$ 2) $U$ is linear and ...
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53 views

Orthonormal bases for Hilbert spaces

In Reed and Simon (Functional Analysis) Theorem II.6 states that, given an orthonormal basis $\{ x_\alpha \}_{\alpha \in A}$ (not necessarily countable)for a Hilbert space $H$, every $y \in H$ can be ...
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165 views

Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
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201 views

Weak* operator topology and finite rank operators

We will say that ${T_i}\subset B(X,Y^*)$ converges to $T$ in W*-operator topology if $T_i(x)\rightarrow T(x)$ in W*-topology of $Y^*$( $\forall y\in Y \langle T_i(x),y\rangle \rightarrow \langle ...
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29 views

About weak-measurability in L(X,Y)

I'm starting a project about vector- (Banach) valued funtions and measures, and I know some of the basic definitions (measurable,weakly measurable...). The functions of study are ...
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1answer
150 views

Adjoint of resolvent of self-adjoint, densely-defined operator on a Hilbert space

Let $H$ be a Hilbert space, $T=T^*$ a densely-defined linear operator on $H$. Denote the resolvent set of $T$ as $\rho(T)=\{\lambda\in\mathbb{C}~|~T-\lambda$ has bounded, everywhere-defined inverse}, ...
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158 views

What is the relationship between spectral resolution and spectral measure?

In Kadison and Ringrose's book "FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS", the author gives the following theorem. Theorem: If $A$ is a self-adjoint operator acting on a Hilbert space ...
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37 views

Composition of analytic functions is analytic in a general setting, and are they continuous?

Regarding the notion of analyticity discussed in this setting: A possible equivalence for holomorphicity I wonder if this is truly the correct definition (even though it is from Dunford-Schwarz) An ...
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63 views

If $X‎^{*}$ has Daugavet property, then X is likewise, but the converse is not valid.

I read in an article that If $X‎^{*}$ has Daugavet property, then X is likewise. I do search its proof in webs and "google scholar or book" but i dont find it. if you can prove it or give me a topic ...
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149 views

Matrix representation of a co-domain restriction of a linear operator

Consider the finite-dimensional linear operator: $\mathcal{A}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3},$ with $Ax=y,$ $A=\left[\begin{array}{ccc} 1 & 0 & 1\\ 1 & -2 & -1\\ 0 & 1 ...
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47 views

Is this operator closed?

Consider the linear operator $H$ with domain $D(H) = S(\mathbb R)\subset L^2(\mathbb R)$, where $S(\mathbb R)$ is Schwartz space, defined by \begin{align} H\psi(x) = -ix^3\frac{d\psi}{dx}(x) -i ...
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140 views

Confusion in Gelfand theorem in C*-algebra.

I am reading HX Lin's book, named "An introduction to the classification of amenable C*-algebras", I can not understand a corollary of Gelfand theorem(Corollary 1.3.6): If a is a normal element in a ...
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26 views

Unbounded family of isomorphisms

Suppose $(T_\alpha)$ is a unbounded in norm family of onto isomorphisms acting on a Banach space. Does it follow that the family $(T_\alpha x)$ is unbounded for any non-zero $x$?
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1answer
136 views

uniform convergence on compact subsets of the linear,continuous and uniformly bounded operators.

Let $X,Y$ be normed spaces. Let $T_j : X\to Y$ be a sequence of linear and continuous functions, such that $\lVert T_j\rVert\lt K$ $\forall j$. If $T_j$ converges pointwise to $T$, prove that $T$ is ...
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44 views

tight frame for $\mathbb{C}^N$

I have a question to ask Prove that if $K\in\mathbb{Z}-\{0\}$, then $\{\phi_p[n]=\exp(i2\pi pn/(KN))\}_{0\leq p<KN}$ is a tight frame of $\mathbb{C}^N$, i.e. $\sum_{k}|\langle f,\phi_p\rangle ...
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45 views

A question about compact Hausdorff space

Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. And $F$ is a closed subspace of $X$. If the $f\in C(X)$ such that $f|_{F}=0$ is only zero function( i.e. ...
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A question about bounded operators on banach space [duplicate]

Let $L(X)$ denotes the Banach algebra of all bounded linear operators acting on a Banach space $X$. And $T$ is not invertible. Can we find a invertilbe bounded operator series $\{T_{n}\}$ such that ...
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A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
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1answer
165 views

Example of non-orthogonal projection on Hilbert space

Can anybody cook up an example of a projection operator $P$ on a Hilbert space $H$ that is non-orthogonal? I.e., one where $PH$ and $(1-P)H$ are not orthogonal subspaces of $H$. I'm completely ...
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28 views

Uniqueness of solution to integral equation for operator valued functions

Let $X$ be a Banach space. Suppose I have a 2 parameter family of bounded operators on $X$: $V(t,s)$, $0\leq s\leq t \leq T$, such that $V(t,s)x=U(t,s)x+\int_s^t V(t,r)H(r)U(r,s)x\,dr$ and ...
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94 views

What is a concrete example of a non-compact Hermitian operator on an infinite-dimensional Hilbert space whose eigenvectors do not form a complete set?

If I am not misunderstanding anything: by the spectral theorem, Hermitian operators that act upon finite-dimensional Hilbert space as well as compact Hermitian operators that act upon ...
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71 views

bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
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80 views

Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is ...
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63 views

How can I better understand manipulating “operators” in mathematical relations?

Sometimes, (especially in physics), it's common to see mathematical relations manipulated and/or derived by separating "operators" from the things they "act on." I can usually keep up with and follow ...
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61 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
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63 views

if T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎

I want to show that if the operator T is normal, then $‎\sigma(T)=‎\sigma‎‎_{‎ap‎}‎(T)‎‎‎‎‎$‎ Its proof is obvious from one hand.But i cant prove that ...
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114 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
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1answer
50 views

A bounded everywhere defined operator that is affiliated to a von Neumann algebra is in the algebra

A possibly unbounded operator $T$ on a Hilbert space $\mathcal H$ is (in my source) defined as affiliated to a von Neumann algebra $M$ if for each unitary element $u$ of $M^\prime$, $u^*Tu=T$ (or ...
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108 views

How would you determine whether this sequence transformation has an inverse?

Let $T : a \mapsto b$ be a transformation of sequence $a$ to $b$ of the form $$ T(a)_m = b_m = \sum_{k=1}^{\infty} a_k e^{-i 2 \pi m / k } $$ Question. How would you go about determining if this ...
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27 views

Pointwise convergence of resolvent

Suppose $T$ is a quasiniplotent operator and $\lambda_n$ a sequnce converging to $0$. Then clearly $||(\lambda_nI-T)^{-1}||\to\infty$. I am interested for which $x$ we have that ...
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1answer
47 views

Conditions necessary for commutators [A,B]=[B,A]?

I know that normally for commutators that [A,B]=-[B,A] where A and B are operators. But under what conditions does [A,B]=[B,A]?
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98 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...