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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Find the norm of an operator on $\ell_2$

Let $(x_n) \subset \ell_2$ and let operator $L:\ell_2\to \mathbb R$ be defined by: $\displaystyle L((x_n)) := \sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n(n+1)}}$. Find the norm of L.
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75 views

How to show $(A^{T}A)^{2}=A^{T}A$, where A is a matrix.

$A$ is a matrix that is $k$ x $n$ and is $\in\mathbb{R}$, where n is greater than k. Also, the set of row vectors is orthonormal w.r.t. dot product. Show $(A^{T}A)^{2}=A^{T}A$. I know bits and ...
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89 views

Diagonal operators on non-separable Hilbert space

Let $H$ be a non-separable Hilbert space with an orthonormal basis $(e_\alpha)_{\alpha<\omega_1}$. To each $f=(f_\alpha)\in c_0(\omega_1)$ associate an operator on $H$ defined by $T_f ...
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1answer
209 views

eigenvalue question

I think this question isn't that hard, but I am a bit confused. Define the linear operator $T_k:H\mapsto H$ by \begin{align} T_ku=\sum^\infty_{n=1}\frac{1}{n^3}\langle u,e_n\rangle e_n+k\langle ...
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0answers
66 views

Positive maps on $\mathcal{B}(\mathcal{H})$ to itself

Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
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1answer
97 views

Properties on Hilbert space

$H$ is real Hilbert space. $a\colon H\times H \to \mathbb R$ is a bilinear form on $H$ with $\lvert a(x,y)\rvert \leq C\lVert x\rVert \lVert y\rVert$ and $a(x,x) \geq \alpha \lVert x\rVert^2$. I would ...
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1answer
173 views

Norm of operator $g\mapsto \int fg$

Let $T_f:(C([a,b],\mathbb C), \lVert \cdot \rVert_1) \to \mathbb C$ with $g\mapsto \int_a^b f(x)g(x) dx$ for any given $f\in C([a,b],\mathbb C)$. I have to find the norm of $T_f$. I started with: ...
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1answer
92 views

analogue of diag operator for functions

If $x\in{\rm I\! R}^n$, then diagonal matrix $\mathop{\rm diag}(x)$ is a linear operator $\mathop{\rm diag}(x): {\rm I\! R}^n \to {\rm I\! R}^n$. I am curious if there is some analogy for infinite ...
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1answer
444 views

Show that $A^{\dagger^{\dagger}} = A $

How do we show that $A^{\dagger^{\dagger}} = A $ without assuming $A$ to be a explicit matrix. That is, given a linear operator $A$, let us define $A^\dagger$ to be a unique operator such that ...
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1answer
180 views

Abelian von Neumann Algebras on non-separable Hilbert spaces

Is there a classification of Abelian von Neumann algebras on non-separable Hilbert spaces? For a classification of Abelian von Neumann algebras on separable Hilbert spaces, see this link.
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104 views

Help with an operator norm

Let $T\in \ell_\infty(\mathbb{Z,\mathbb{C}})^*$ such that: $T(1_{\ell_\infty})=1$ where $1_{\ell_\infty}$ denotes the constant function $1$; $T(u)\geq 0$ whenever $u$ is real positive. How to ...
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1answer
164 views

Reducing subspaces for compact operators

It is well known that any compact operator in $\mathcal{B}(l_2)$ has an invariant subspace. What about reducing subspaces (subspaces that are invariant for both the operator and its adjoint). Does any ...
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1answer
58 views

Injectivity of a certain operator

Consider a compact $K$ of the complex plane of interior void, $f$ a rational fraction leaving $K$ invariant (i.e. $f(K) \subset K$), and $\mu$ a borelian probability measure supported by $K$, and ...
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1answer
76 views

Banach space geometry without bounded operators?

I understand that $B(X)$ can be think of as the collection of symmetries of a Banach space $X$, and that they provide important information concerning the geometric structure of the space. But I am ...
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1answer
113 views

Positive operator is bounded

For a real Banach space $X$ let $A:X\rightarrow X^*$ be a positive operator in the sense that $(Ax)(x)\geq 0$ for all $x\in X$. Show that $A$ is bounded. I don't know how to do that, maybe it's ...
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394 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
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2answers
188 views

Compact and self-adjoint operator

It is true that if $T:H \to H$ is a compact operator ($H$ Hilbert space) then $T^\ast T$ is algo compact and indeed self-adjoint. Conversely, is it true that every compact and self-adjoint operator ...
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1answer
112 views

Characterizing positive semi-definite operators in $\mathcal{B}(L^2)$

I am asking perhaps a stupid question. How can I characterize all positive semi-definite operators in $\mathcal{B}(L^2(X,\lambda))$, where $\lambda$ is the Lebesgue measure. For a start, let us ...
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1answer
79 views

Pseudo Monotone Operator

Suppose $X$ is a real Reflexive Banach space. Let $A:X\rightarrow X^{\star}$ be a Pseudo Monotone operator, i.e. if $u_{n}\rightharpoonup u$ and $\limsup\langle Au_{n},u_{n}-u\rangle\leq 0$, then ...
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1answer
363 views

In a separable Hilbert space, how to show that the orthogonal projection onto a subspace of $n$ orthonormal basis elements converge?

Could anyone help me with this problem? I don't know where to start. Let $\{ e_n \}_{n=1}^\infty$ be an orthonormal basis in a separable Hilbert space $H$. Denote by $P_n$ the orthogonal ...
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1answer
801 views

Norm of integral operator in $L^1$

What is the norm of the operator $$ T\colon L^1[0,1] \to L^1[0,1]: f\mapsto \left(t\mapsto \int_0^t f(s)ds\right) $$ ?
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98 views

Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
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1answer
205 views

Extension of Choi's theorem on extreme completely positive maps

In this paper Man-Duen Choi gave a criteria for a completely positive map to be extreme. For convenience I am writing it below. Let $\phi:\mathcal{M}_n\rightarrow\mathcal{M}_m$. Then $\phi$ is ...
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2k views

An inequality on trace of product of two matrices

Suppose we have two positive semi-definite matrices of dimension n, $A$ and $B$ s.t. Tr$(A)$, Tr$(B)\le1$. Can we say anything about Tr$(AB)$? (Is Tr$(AB)\le1$ too?)
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1answer
71 views

Question about locality of bounded linear operators between Banach spaces

Suppose $T$ is an invertible bounded linear operator from $X$ to $Y$, which are Banach spaces. Is there a neighborhood $U$ of $T$ in $\mathcal{B}(X,Y)$ (space of bounded linear operators from $X$ to ...
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0answers
39 views

Initial topology of the spectrum mapping $\sigma$

Let $\mathcal{A}$ be a Banach algebra, the map $\sigma$ maps each element $a\in\mathcal{A}$ to its spectrum $\sigma(a)$, which is a compact subset of $\mathbb{C}$. The collection of compact subsets ...
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1answer
150 views

Strong continuity of the Borel functional calculus

I have sometimes heard that the Borel functional calculus maps bounded pointwise convergent sequences of Borel functions to strongly convergent sequences of operators. I gather "sequence" is ...
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1answer
588 views

Commutator relationship proof $[A,B^2] = 2B[A,B]$

I'm trying to find the condition necessary for this commutator relationship equality: $$[A,B^2]=2B[A,B]$$ So far I've done this: \begin{align*} [A,B^2] & = B[A,B] + [A,B]B \\ ...
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1answer
203 views

A characterization of trace class operators

Let $H$ be a separable Hilbert space and let $T\in B(H)$ be compact (I don't know whether this is relevant to the question), such that $\displaystyle \sum_{j=1}^\infty\langle T\xi_j,\eta_j\rangle$ ...
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1answer
249 views

Equivalence of Schatten and spectral norms

I'd like some help showing the equivalence of these two norms when $p = \log n$. Recall the $p$-th Schatten norm of a linear operator $A$ acting on $\mathbb{R}^{n}$. In the particular case of $p = ...
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1answer
51 views

Conditions for additivity of the trace of projections

I came up with the following problem: Let $\Pi_A$ and $\Pi_B$ be two projection operators on two disjoint subspaces of a certain Hilbert space $\mathcal H$ and let $\rho$ be unit trace, positive, ...
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1answer
231 views

Unique extension to a bounded operator

Suppose $\left\{ e_{1},e_{2},\ldots\right\} $ is an orthonormal basis for a Hilbert space $\mathcal{H}$ and for each $n$ there is a vector $Ae_{n}$ in $\mathcal{H}$ such that $\sum\left\Vert ...
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1answer
132 views

Finding a linear mapping in a special Hilbert space

Let $H=\ell_2$, the real Hilbert space whose elements are the square-summable sequences of real scalars, i.e., $$ H=\left\{u=(u_1,u_2,\ldots,u_i,\ldots): ...
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1answer
283 views

Spectrum of this Operator

Let $A\colon \ell^{1}\to \ell^{1}$ be defined by $A(x)=(x_{2}+x_{3}+x_{4}+ \dots,x_1,x_2,x_3,\dots)$ where $x\in\ell^1$ iff $\sum|x_k|<\infty$. Let $D$ be the closed unit disc in $\Bbb C$ and ...
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55 views

On Scalar products — What is wrong with this argument?

Suppose $|k\rangle =\exp(k \hat O^\dagger)|0\rangle$ where $c_k\in \mathbb C$ and $|0\rangle$ is normalized. I wish to evaluate $\langle a|b\rangle$. Here is what I think, but the result is not ...
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1answer
286 views

Reflexivity of a Banach space

I've run into a few problems in which reflexivity of a Banach space is given as a hypothesis. These problems are sometimes of the type where the banach space is specific/concrete, and sometimes it is ...
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3answers
626 views

operator exponential

The matrix exponential is a well know thing but when I see online it is provided for matrices. Does it the same expansion for a linear operator? That is if $A$ is a linear operator then ...
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1answer
119 views

Weak analyticity vs. Strong Analyticity

Let $X$ be a (complex) banach space, $U$ be an open subset of $\mathbb{C}$ and $f: U \to X$ be a function that is completely arbitrary except that it satisfies the property that for any continuous ...
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1answer
144 views

What is an OPERATOR?

I try to understand operator-valued kernels. For this purpose, first want to know what is an operator. I can see the definition of operator here, but I do not quit get it. Can anyone explain it in ...
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2answers
103 views

Multiplication operators

Consider a commutative Banach algebra $A$ and the Banach algebra of bounded operators $B(A)$ on $A$. Associate to each $a\in A$ the multiplication operator $T_ax =ax$ ($x\in A$). Is always the mapping ...
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186 views

Behaviour of the spectrum of a compact operator w.r.t. perturbations.

Suppose $A$ and $B$ are linear compact operators on a Hilbert space with $\sigma(A)$ and $\sigma(B)$ as their spectrum. Is it possible to obtain some continuity result of $\sigma(A+\epsilon B)$ as ...
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1answer
582 views

The relation between bounded invertible and surjective operators

Please, answer me that how is the set of all bounded invertible operators (for example on a Hilbert space) clopen (closed and open) in the set of all bounded surjective operators? In fact, which ...
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1answer
450 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
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1answer
170 views

Fredholm operators

How can I get the (Volterra) operator from an equation of the type $$u''(x)+xu'(x)+u(x)=0\text{ ?}$$ I know that there is a general way of doing it, if you could point me at the proper book I'd be ...
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1answer
254 views

Error in proof of self-adjointness of 1D Laplacian

I have successfully checked self-adjointness of simple and classic differential operator - 1D Laplacian $$D = \frac {d^2}{dx^2}: L_2(0,\infty) \rightarrow L_2(0,\infty)$$ defined on $$\{f(x) | f'' ...
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116 views

Composition of positive maps

Let $\chi_A:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ be a completely positive (cp) map defined as $\chi_A(x)=AxA^*$, where $A\in\mathcal{B}(\mathbb{C}^n)$. Clearly any cp map ...
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117 views

Set of Bounded linear Operators on $l_2$ is dense on the set of bounded operators on $l_2$?

Let $l_2^{+}$ be the Hilbert space of all square summable sequences $\{x_n\}, n \in \mathbb{N}$ under some definition of inner-product $\langle,\rangle_l$. Define $B[l_2^{+}]$ as the set of all ...
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2answers
722 views

The spectrum of an unbounded operator

It's well known that the spectrum of a bounded operator on a Banach space is a closed bounded set (and non-empty)on the complex plane. And it's also not hard to find unbounded operators which their ...
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217 views

Help for Divergence operator

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple. Can some one tell me some reference to study about the invertibility of Divergence operator ...
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381 views

Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem Does the following generalization of that fact also hold? Theorem: ...