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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Norm of integral operator

Consider the operator $T(f(t)) = \int_0^t f(s)ds$, where $t \in [0,1]$, and $f(t) \in C[0,1]$. To prove $$\|T^n\| = \frac{1}{n!}$$ Thanks for suggestions.
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similarity between bundle shift

Let $E$ be a flat unitary bundle of rank $n$ over a domain $R$ in $\mathbb{C}$. It is known that bundle shift $T_{E}$ is similar to $T_{\mathbb{C^n}}$ (which is the bundle shift corresponding to the ...
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Matrix form of the differential operator $\sum_{k=1}^N x^k\frac{d^k}{dx^k}$

The following differential operator: $P(x,N)=\sum_{k=1}^N x^k\frac{d^k}{dx^k}$ is defined in $x\in\left[-1,+1\right]$. Is it possible to find a matrix form of this operator vs. $N$? Because it's ...
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Spectrum of linear operators

I can't solve the following: i) Let $T:l^2 \rightarrow l^2$ , $Tx=\{ (Tx)_n\}_{n=1}^{\infty}$ given by $$(Tx)_n = \dfrac{1}{2}x_{n-1} + \dfrac{1}{2}x_n.$$ Find $\sigma(T)$. ii) Let $S : l^2 ...
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247 views

Simple spectrum and the spectral theorem for bounded symmetric operators

I have a question regarding the spectral theorem for bounded self-adjoint operators. The book "Functional Analysis, an Introduction" by Eidelman, Milman, and Tsolomitis says that if an operator $T$ ...
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Strong convergence of operators

I'm working through the functional analysis book by Milman, Eidelman, and Tsolomitis, and I have a question. The book states a lemma that I'm a bit confused about: A sequence of operators $T_n\in ...
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417 views

Decomposition an operator in terms of symmetric and anti-symmetric components

In linear algebra, we can write any operator as the sum of a symmetric and skew-symmetric parts: $$A=A^{\mathrm{sym}}+A^{\mathrm{skew}}$$ where $$A^{\mathrm{sym}}=\frac{1}{2}(A-A^T)$$ and ...
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A theorem about operator theory

Define $$\operatorname{Ref}\mathcal{S}=\{T\in B(\mathcal{H}):Th\in[\mathcal{S}h], \forall h \in \mathcal{H}\},$$where $\mathcal{H}$ is a Hilbert space and $\mathcal{S}$ is a linear manifold of ...
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Self-adjoint extensions modern paper or book

Do you know some modern and recent paper, lecture notes, or book about self-adjoint extension theory (defect indeces, Von Neumann theory,...)? Classical references can also be helpful but I am ...
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An inequality involving operator and trace norms

Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...
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Core for the Laplace operator in a bounded domain

Let $X$ be a bounded connected open subset of the $n$-dimensional real Euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...
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137 views

approximation property

In I. Namioka and R. R. Phelps's your paper "Tensor products of compact convex sets" Pacific Journal of Mathematics, Vol. 31, No. 2, 1969), they gave the following definition of approximation ...
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79 views

Calculating the norm of $S\otimes T$ for bounded linear $S,T$.

Let $X,Y,W,Z$ be Banach Spaces. Let $X\otimes_{\pi}Y$ denote the tensor product endowed with the projective norm. If I have $S\in B(X,Z)$, $T\in B(Y,W)$, it is straightforward to show that $S\otimes ...
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167 views

Quadratic Operator Notation?

I am dealing with functions that are linear combinations of: $[x_1^2, x_2^2... x_n^2, x_1x_2, x_1x_3... x_n-1x_n]$ spanned over a column. All these functions obey the law: $F(aX) = a^2F(X)$ for ...
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108 views

Function of a completely continuous operator

I would be most thankful if you could help me with this question. If $A$ is a completely continuous Hermitian operator on a Hilbert space $H$, for what class of functions $f$ can one define a function ...
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181 views

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just ...
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132 views

how to prove this property of compact operator? [duplicate]

I read about this property of compact operator from wikipedia $K(X, Y)$ is a closed subspace of $B(X, Y)$: Let $T_{n}, n \in N$, be a sequence of compact operators from one Banach space to the other, ...
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50 views

Question about the terms and operations in basic division

Let's pretend that I am a child and you want to teach me division. You demonstrate through an example division as repeated subtraction. This is the simple algorithm the child learns from your ...
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53 views

Is there an asymmetric positive definite second-order linear differential operator?

The second-order differential operator is $Lu=-\sum_{i,j=1}^n (a^{ij}(x)u_{x_i})_{x_j} +\sum_{i=1}^n b^i(x) u_{x_i} +c(x) u$. We say it's positive definite if there exsits constant $\beta>0$ such ...
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Sufficient condition for self-adjoint subset of bounded linear operators on a Hilbert space being irreducible

Let $H$ be a Hilbert space and denote as $B(H)$ the bounded linear operators on $H$. Let $M$ be a subset of $B(H)$, s.t. for $A \in M$, also $A^* \in M$. How can one show that if the commutant has ...
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113 views

Operator identity involving square root of an operator

I would be most thankful if you could help me prove the following identity. Let $A$ and $B$ be two completely continuous Hermitian operators on a Hilbert space $H$, such that $A$ and $B$ do not ...
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111 views

Self-adjoint operator on a finite-dimensional vector space

Let $V$ be a finite-dimensional inner product space and let $x,y\in V$ be nonzero vectors. If there is a self-adjoint operator $A:V\rightarrow V$ such that $A(x)=y$ and $\langle A(v),v\rangle\geq0$ ...
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389 views

Proof Complex positive definite => self-adjoint

I am looking for a proof of the theorem that says: A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
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application of the theorem of the open application

Let $ X, Y $ be Banach spaces. Suppose that $ T: X \to Y $ is a compact operator. show that if $ \dim Y $ is infinite, then $ T $ is not surjective. idea: Using the theorem of the open application
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Relationship of two generalizations of the real/complex calculus

On the one hand, one has the various functional calculi from Operator Algebras. The continuous functional calculus for C* algebras, the bounded borel functional calculus for Von Neumann Algebras, the ...
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Elliptic Operators

I'm studying Elliptics Operators like this: $$Lu=a_{ij}(x)D_{ij}u+b_i(x)D_iu+c(x)u$$ for $u\in C^2(\Omega)\cap C(\overline{\Omega})$. I want to know what the difference when: $L$ is elliptic in ...
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Adjoints of an operator

In a paper I'm reading the discussion centres on operators of the form \begin{equation} \mathbf{B} = (-1)^{m+1} \Delta^m_y + \frac{1}{2m}y\cdot\nabla_y \end{equation} Apparently this is symmetric ...
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90 views

Core of multiplication operator

Suppose $\phi$ is a continuous function on $\mathbb{R}$. Suppose we have an operator on $L^{2}(\mathbb{R})$ defined by: $$ D(T) = \{ f \in L^{2}(\mathbb{R}) \; | \; f\cdot \phi \in L^{2}(\mathbb{R}) ...
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Theorem of Lidijski

I'm sorry this question is so "small". My Operator Theory professor said we'd be proving The theorem of Lidijski next week. So I googled it and found nothing, and then i sort of panicked. So that is ...
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Is this operator bounded? Hilbert space projection

Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense in $H$. Let $b_n$ be an orthonormal basis for $H$ and an orthogonal basis for $V$. Define $$P_n:H \to ...
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Is the Laplace operator in Sobolev space $W^{2,1}$ sectorial?

Let $\Delta$ be the Laplace operator, that is $\Delta f = f''$. It is well known (see, e.g. this lecture notes, Chapter 2) that $\Delta\colon D(\Delta) \to L^1(0,1)$ with domain $D(\Delta) = ...
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About the Spectrum of operators

I'm studying operator theory, and a doubt come at me, we know the diference between the pontual spectrum, the continuous spectrum and the residual spectrum. And we have that $\lambda \in \sigma(T)$ ...
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Proof that restriction of hermitian operator to its invariant subspace is also hermitian

Proof that restriction of hermitian operator to its invariant subspace is also hermitian What would be the most elegant way to prove this?
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taylor series for a function of matrices

Say I have a function $(A+B)^{-1}$ where $A$, $B$ are matrix-valued functions of some vector $x$. Can I then expand this function around $x=0$ as: $$(A+B)^{-1} = (A[0]+B[0])^{-1} - (A[0]+B[0])^{-2} ...
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Prove operator $T$ is onto

Consider the Hilbert spaces $X := H^{1}(\Omega)\times H^{1}(\Omega)$ and $Y:=L^2(\Omega)\times L^2(\Omega)$, where $\Omega =\ ]{-}\pi, \pi[$, and \begin{eqnarray*} \langle(u,v), (z,w)\rangle_X & = ...
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Operators from $L^{\infty}$ to $L^{\infty}$, below bound of the norm

If $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded on $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le \|T\|_{L^{\infty}\rightarrow ...
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Compact integral operator

I have this exercise and I don't know how to solve the last question. In the following $a,b$ are two real numbers such that $a<b$ ,$E=C([a,b],\mathbb{R})$ with the norm $||.||_0$ given by ...
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299 views

A relatively bounded perturbation of a closed operator is a closed operator.

Please I need help with an example I cant figure out and which will hopefully help me to get the theory: Let $X$ be Banach space and $A, B$ general operators. Furthermore $A$ is closed, ...
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115 views

Operator on $\ell^2$

I have an operator $A \in B(\ell^2)$, $Ax=(0,x_1,\frac{1}{2}x_2, \frac{1}{3}x_3,...),\; x=(x_1, x_2, x_3,...) \in \ell^2$. I need to find the following and I am not sure even how to start: proof ...
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267 views

Bounded operator inverse, norm and spectrum

I need help with an operator, I am not very good at functional analysis and need to find some properties of following operator: $X=C[(0,1)], A \in B(X); A[f(t)]=f(t^2)$ 1. I need to show that an ...
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Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
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not injective/not surjective linear maps

Let $S$ be the vector space of real sequences, and for $x=(x_1,x_2,\dots)$ define $\alpha(x)=(0,x_1,x_2,\dots)$ and $\beta(x)=(x_2,x_3,\dots)$. The problem was asking for few other things to do, but I ...
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An operator inequality

I would be most thankful if you could help me prove the following operator inequality. Let $A$ be an arbitrary linear operator on a Hilbert space, satisfying $$\left\|AA^{\ast} - A^{\ast}A\right\|\leq ...
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310 views

Finding operator norm

I have to solve the following problem: Find a norm of operator $$A:L^2[-\pi,\pi]\rightarrow L^2[-\pi,\pi]$$ given with $$Af(x)=\int_{-\pi}^{\pi} \cos^2{\left(\frac{x-t}{2}\right)}f(t) \,dt.$$ I ...
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A question concerning the Hilbert space trace

I am stuck with an equation regarding the trace on a Hilbert space $H$. The trace is defined in the book by Pedersen ("Analysis now", Sect. 3.4) as follows. We choose an orthonormal basis $\{ e_j ...
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55 views

Operator inequality

I would be most thankful if you could help me prove the following inequality. Let $A$ be a linear Hermitian positive operator on a Hilbert space. Then show that $$\langle x,A^{2}x\rangle\geq \langle ...
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Can I always extend a selfadjoint Operator in $L^2$?

Assume that we have a self-adjoint operator $T\colon D \to D$ where $D \subset L^2$ is some finite dimensional subspace. Can I conclude that than a self-adjoint operator $S \colon L^2 \to L^2$ exists ...
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Prove that there $B : H\to H $ bounded such $ B^n = A $.

Let $ A : H\to H $ a compact self-adjoint operator. Suppose $ A $ is positive. let $ n \geq 2 $. Prove that there is $B : H\to H $ bounded such $ B^n = A $.
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How to prove that $i_{k,\ k+1}$ it is compact?

Let $i_{k,\ k+1}$: $H^k(\mathbb{T}^d) \to H^{k+1}(\mathbb{T}^d)$ a identy operator where $H^k(\mathbb{T}^d)$ it is Sobolev_space in toro $\mathbb{T}^d$. Prove that $i_{k,\ k+1}$ it is compact ...
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On the use of nets when defining operator topologies

Let's consider the strong operator topology and the weak operator topology on bounded operators of a infinite-dimensional Hilbert space $H$. When they define these operator topologies, some authors ...