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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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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119 views

Does ternary operations have associative property?

Binary Operation is a function. Right? We know that all Binary operations have associative property. They must be either associative or non-associative. The condition is : $$(a*b)*c = a*(b*c)$$ ...
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151 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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1answer
58 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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64 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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3answers
65 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: ...
2
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2answers
58 views

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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1answer
54 views

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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1answer
242 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 ...
2
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1answer
57 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 $T_2^*X=XT_1^*$. ...
4
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68 views

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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100 views

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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1answer
210 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 ...
3
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1answer
77 views

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 ...
3
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1answer
49 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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625 views

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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3answers
2k views

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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20 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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6answers
437 views

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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2answers
67 views

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$

$T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$ 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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171 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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86 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
63 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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1answer
61 views

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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1answer
118 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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1answer
53 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 ...
4
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1answer
119 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
96 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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78 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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108 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 ...
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1answer
238 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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186 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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45 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 ...
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1answer
61 views

Intuition concerning Schwartz kernels of Operators

Consider a (for example differential) operator $A$ acting on an appropriate function space over a smooth compact manifold without boundary. Using the Schwartz kernel $K(x,y)dy$ of the operator, its ...
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83 views

Transpose of the Hilbert-Schmidt operator

Let $X = L^2(\Omega)$, $\Omega \subset \mathbb{R}^N$ be an open set (or a $\sigma$-finite measure space), $B \in L^2( \Omega \times \Omega)$. Then the Hilbert-Schmidt operator $T \in \mathcal L(X)$ ...
3
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60 views

$B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuos operator

Let $B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuous operator such that $T(B)=B$ and $T(x)=0\Rightarrow x=0$ which of the following is correct? $T$ maps bounded sets into ...
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63 views

Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
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116 views

Counterexample using counting measure

While proving that the norm of the mulplicative operator from $L^2(X) \to L^2(X)$ is the essential supremum of $|g|$ where $g \in L^\infty(X)$, I found that I need the $\sigma$-finiteness of the ...
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0answers
40 views

self adjoint linear operator and integration

is this formula correct ?? $$ \int_{-\infty}^{\infty} Lf(x)\delta (x-1)= \int_{-\infty}^{\infty} f(x)L^{\dagger}\delta(x-1) $$ here $ L $ is a linear operator and $ L^{\dagger}$ is its formal ...
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301 views

Shift Operator has no “square root”?

Consider the left shift operator $T : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $ by $$T(x_1,x_2..... )=(x_2, x_3 ........),$$ and also the right shift operator $S : \ell^1(\mathbb N) \to \ell^1(\mathbb ...
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1answer
72 views

Left support of an operator on a Hilbert space

The left support $l(x)$ of an operator $x$ between Hilbert spaces $\mathbb{H}$ and $\mathbb{K}$ is defined as the smallest projection $e \in \mathfrak{B}(\mathbb{H})$ such that $ex=x$. The question ...
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1answer
40 views

a question on decreasing sequence of subspaces II

This is related to this question see here Let $V= \mathbb{Q}^{\mathbb{Z}}$, considered as an infinite dimensional linear space over $\mathbb{Q}$. And assume $W=\mathbb{Q}^F$ is a finite dimensional ...
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1answer
76 views

Reflexivity of $X \times Y$

I want to prove the following Theorem. Let $X,Y$ be reflexive. Then $X \times Y$ is reflexive. Here my try. Proof. Let $J_X, J_Y$ be the canonical injections of $X$ onto $X''$ and of $Y$ onto ...
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1answer
48 views

Help showing $\phi _k$ is a bounded linear functional

Let $V$ be the space of continuous functions on the interval $[-\pi , \pi]$ with the $L^2$ norm $$\lVert f\rVert_2=\left(\int_{-\pi}^\pi |f(t)|^2\mathrm dt)\right)^\frac{1}{2}$$ For $f$ in $V$, define ...
2
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1answer
64 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
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58 views

Almost everywhere analytic function

Suppose we have a measure space $\Omega$ and a function $m\in L^\infty(\Omega,\mathcal{B}(E))$, that is invertible for almost all $\theta\in\Omega$ Further assume, that we have an other function $G$ ...
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1answer
185 views

Adjoint of a multiplication operator

Let $B$ be the Banach space of continuous functions vanishing at infinity and defined on a locally compact Hausdorff space $X$. Given a continuous and bounded function $g$ on $X$, let $T$ be the ...
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2answers
74 views

$a^*a$ has a non-negative spectrum

I am learning $C^*$-algebra, especially I work on the proof of the Gelfand-Naimark theorem. In many books such as the one of Arveson, it looks that the following lemma is the key stone of the proof: ...
2
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1answer
155 views

Show that an operator is bounded (from Reed and Simon)

I am currently reading Reed and Simon's IV: Analysis of Operators, Volume 4 (Methods of Modern Mathematical Physics). I don't understand something they do in Theorem XIII.64. The problem is: Let $A$ ...
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1answer
120 views

A problem with linear operator in a Hilbert space

Let $(H,(\cdot,\cdot)_H)$ and $(Q,(\cdot,\cdot)_Q)$ two Hilbert separable spaces s.t $H\subset Q$ and let $B:H\to Q$ a bounded and linear operator. Let $\sigma,\tau\in H$ two fixed elements. My ...