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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Differential operators: elliptic vs strongly elliptic

This morning a collegue of mine came to me with the following question: does there exist any elliptic operator of order $2m$ with real (variable) coefficients that is not strongly elliptic? After ...
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349 views

Spectrum and point spectrum of this operator

Let $T\in \text{Aut}(\ell^2(\mathbb{C}))$ and $T(x)=(a_1 x_1, a_2 x_2,\ldots)$ where $a=(a_i)_i \in \ell^\infty(\mathbb{C})$. How can I easily see what is $\sigma(T)$ and $\sigma_p(T)$ (that are ...
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82 views

Computing $e^{isD}$ for a differential operator D

I'm trying to understand functional calculus of unbounded operators and everywhere I see proofs of its existence, but it seems that no one ever dares to compute some easy example. Lets take $D = ...
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2answers
215 views

operator norm of this multiplier operator

I am having some trouble with some basic properties of a given operator. Firstly, the operator T is defined as taking the fourier inverse transform of the function ...
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63 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
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534 views

Commuting operators and polar decomposition

Suppose that $V$ is an isometry and $X$ an arbitrary operator on a Hilbert space $H$. Let $X=U|X|$ be the polar decomposition for $X$. If $VX=XV$, can I conclude that $VU=UV$?
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534 views

No Nonzero multiplication operator is compact [duplicate]

Let $f,g \in L^2[0,1]$, multiplication operator $M_g:L^2[0,1] \rightarrow L^2[0,1]$ is defined by $M_g(f(x))=g(x)f(x)$. Would you help me to prove that no nonzero multiplication operator on $L^2[0,1]$ ...
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35 views

Is this argument valid?

Can someone check if the following argument is valid? Show that if $T$ is an operator and operator $T \circ T$ has eigenvalue $\lambda^{2}$, than $\lambda$ or $-\lambda$ is an eigenvalue of $T$. ...
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51 views

Does every operator have a matrix?

Maybe this question is basic however I'm not familiar with operator theory. Does every operator have a matrix? I also would like to see some proof of this fact (if it's elementary) or at least get ...
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187 views

Finding the norm of the operators

How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$? 1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with $$(T_zx)(n)=z(n)\cdot ...
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281 views

For normal $T\in\mathcal{B}(H)$ operator $T$ is injective iff its image is dense

Let $H$ a Hilbert space, $T \in \mathcal{B}(H)$ is normal. Show that: $T$ is injective iff $\mathrm{Im}(T)$ is dense in $H$ Any help is appreciated!
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192 views

Exponential operator on a Hilbert space

Let $T$ be a linear operator from $H$ to itself. If we define $\exp(T)=\sum_{n=0}^\infty \frac{T^n}{n!}$ then how do we prove the function $f(\lambda)=exp(\lambda T)$ for $\lambda\in\mathbb{C}$ is ...
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200 views

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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70 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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76 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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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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64 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
92 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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158 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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79 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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360 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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159 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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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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147 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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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
75 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
109 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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354 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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178 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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105 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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72 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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329 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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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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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
186 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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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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68 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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37 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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124 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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462 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
189 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
216 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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50 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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213 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
130 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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255 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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54 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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257 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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505 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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108 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 ...