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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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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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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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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55 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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A simple question about operator norm

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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52 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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48 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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53 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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90 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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38 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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102 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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25 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
40 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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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 = ...
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Question about convergence in weak operator topology (from Reed and Simon)

I am reading over Chapter VI in Simon and Reed's Functional Analysis. In the first section, the discussion covers various topologies defined on $\mathcal{L}(X,Y)$, the space of bounded linear ...
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26 views

Definition of a norm infinity

I have $u:\mathbb{R}^3\times(0,\infty)\longrightarrow\mathbb{R}$ and $g:\mathbb{R}^3\longrightarrow\mathbb{R}$. Which means the following?: $\|u(\cdot,t)\|_{L^\infty(\mathbb{R}^3)}$, and ...
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27 views

Condition on spectrum of T

Let $T$ $\in \mathfrak{B}(\mathbb{H})$ be normal. Let $A$ be the closed subalgebra generated by $T$, $T^{*}$ and $I$. Suppose $T$ can be approximated in norm by finite linear combinations of ...
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78 views

Does ternary operations have associative property?

We know that all Binary operations have associative property. They must be either associative or non-associative. like "+" operation in general case, when operands are real numbers is associative. a ...
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80 views

Comparison of Strong OPerator and Weak * Topologies on B(H)

It is known that in $\mathfrak{B}(\mathbb{H})$, the weak operator topology (WOT) is contained in both the strong operator topology (SOT) and $\sigma$-weak topology. In general the SOT and the ...
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56 views

A question about download the recent paper. [closed]

I am interested in the journal about operator theory, such as Studia Math and Operators and Matrices. However, my college do not buy some journals. How can I get the paper from these journal?
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1answer
47 views

differentiability/holomorphicity of family of bounded operators

Edit: It seems I made a mistake in the statements on differentiability. I will replace weak differentiable implies strong differentiable with weak continuously differentiable implies strongly ...
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57 views

Operator attains norm

We have the following linear and well-defined mapping $T_{a}(b):=\sum_{j=1}^{\infty}a_{j}b_{j}$, with $\{a_{j}\}\in \ell^{1}$ and $\{b_{j}\} \in c_{0}$. Show that $\|T_{a}\|=\|a\|_{1}$. My work: ...
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What restrictions are there on explicit equations?

So I've always been told that for a function to be considered explicit it can only have one specific output for each input or simply pass the vertical line test. While I can accept that on it's face I ...
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Operators that are not represented as matrices , operating on matrices.

I am currently going through "Log-gases and random matrices" by PJ Forrester. I'm coming from a totally different academic background, and I cannot understand a point of his notation. More precisely, ...
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216 views

An alternate proof of Fuglede's theorem

To prove Fuglede's Theorem for normal operators on a separable Hilbert space, why does it suffice to show that $E(S_1)T E(S_2)=0$ for all disjoint Borel sets $S_1$ and $S_2$, where $E$ is the spectral ...
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52 views

Corollary to Putnam's theorem

Suppose $T_1$ and $T_2$ are normal operators on Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively. Putnam showed that if $X$ is an operator satisfying $T_2X=XT_1$, then $T_2^*X=XT_1^*$. ...
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Given a vector x, can we say something about an A such that A x = x?

Let us assume that a vector $x \in \mathbb{R}^n$ is given and we are looking for a matrix $A \in \mathbb{R}^{n\times n}$ which yields $A x = x$. That is, we perform a sort of reverse questioning: ...
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Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
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135 views

Self adjointness of square root operator

Theorem: If $A$ is self adjoint and nonnegative, then $A$ has a unique nonnegative square root $A^{\frac{1}{2}}$. As I understand, thesis of this theorem say only about the existence of ...
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Is $B - B'$ self-adjoint provided $B,B'$ are positive operators?

If I have two positive operators $B,B'$ on an arbitrary Hilbert space $\mathcal{H}$ not necessarily over $\mathbb{C}$, how do I know that $B - B'$ is self adjoint? EDIT: Reed and Simon define ...
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45 views

When does an operator commute with another operator given by a series?

Suppose $B$ is a bounded operator on some Hilbert space $\mathcal{H}$, given by a series of the form $$ B = I + \sum^\infty_{k = 1} c_k(I - A)^k $$ where $A$ is a given bounded operator on ...
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Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
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Commutator of $x$ and $p^2$

I have a question: If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is: $[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$ But ...
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19 views

Difference operator endomorphism

Let $\delta : R_{p}[x] \to R_{p}[X] $ the endomorphism of $R_{p}[X]$ such that : $\delta(P(X)) = P(X + 1) - P(X)$ , what is the kernel of $\delta$ ? (i tried to compute it explicitly but that was a ...
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Nilpotent linear operators

Suppose that $T : V \to V$ is a linear operator on an $n$-dimensional vector space $V$. (a) Show that for all $i$, $\ker T^i \subset \ker T^{i+1}$. (b) Show that if $\ker T^k = \ker ...
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$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$ I need to know whether it is self adjoint and unitary operator given that $x_i\in\mathbb C$ I am not able to do it please tell me how ...
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107 views

Is it a unitary, self adjoint and normal operator?

Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know ...
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81 views

Show that $(D_t t)$ is an isomorphism.

Let $B$ be a Banach space, $\epsilon > 0$, and $$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$ Denote $\partial/\partial t$ by ...
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1answer
56 views

A question about quotient space of $R(T^{n})$

I am reading a paper about spectral theory. The author says it is easy to see the following proposition: For $T\in L(X)$, if dim$(R(T^{d})/R(T^{d+1}))<\infty$, then $R(T^{d})$ is closed if and ...
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Are $T,T^2$ compact operators?

$T:l_2\to l_2$ is defined by $T(x_1,x_2,\dots)=(0,x_1,0,x_3,0,x_5,\dots)$ we need to find whether $T, T^2$ are compact or not. I see here the definition of compact operator but I'm not able to apply. ...
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112 views

Can 0 be an eigenvalue?

Let $-\Delta $ be the positive Laplacian and consider the operator $$ -\Delta + V $$ on $L^2(\mathbb{R}^3)$ with domain the Sobolev space $W^{2,2}(\mathbb{R}^3)$. Here $V:\mathbb{R}^3\to \mathbb{R}$ ...
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47 views

Question about operators on Hilbert space

Let $\cal{H}$ be a Hilbert space, $P_1,P_2,\cdots,P_m$ a sequence of orthonormal projections such that $P_iP_j=0$ for $i\neq j$ and $P_1+P_2+\cdots+P_m=I$. Then $\|\sum^m_{k=1}P_kTP_k\|\leq\|T\|$ for ...
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67 views

Decomposing operators into the sum of a quasinilpotent and something else

I seem to remember some result of the following sort: Alleged Theorem. Every bounded operator on a separable complex Hilbert space can be decomposed as the sum of a normal operator and a ...
2
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1answer
69 views

An quasi-nilpotent operator restricted to a subspace is a nilpotent?

I am reading a paper about operator theory, there is a proposition I could not understand. Let $T\in L(X)$ be a quasi-nilpotent operator and $X_{1}$ be a non-zero finite-dimensional subspace of X, ...
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106 views

General Lax-Milgram problem

I'm trying to solve the problem below, part 1. is proved by a simple application of the Lax-Milgram Lemma (General Case, i.e, with inf-sup conditions), in part 3. I think I know how do it, but part 2. ...
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65 views

Maximum of two positive operators

Let $A,B$ be two positive operators in $B(H)$. Does there exist, in general, an operator $C$ such that for each $T$, if $A \leq T$ and $B \leq T$, then $$A\leq C \leq T\quad \text{and}\quad B\leq ...
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103 views

Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
4
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187 views

How to determine a operator norm?

How to solve following: In Hilbert space $W_2^1=\{f:[0,1]\rightarrow \mathbb{C}|f\in AC[0,1], f'\in L^2[0,1]\}$ with scalar product $(f,g)=\int_0^1 f\overline{g}dx+\int_0^1 f'\overline{g'}dx$ is ...
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113 views

Operator Theory References and Topics

I wish to do a reading course in Operator Theory. Thus, I am looking for some references in the area. Right now, I have the following two sources available: Unbounded Self-Adjoint Operators ...
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42 views

When does Gelfand Naimark Theorem Hold?

I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras. In proving that each character has norm equal to $1$, we used the fact that $||e|| = 1$ where ...