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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Prob. 15, Sec. 3.10 in Kreyszig's functional analysis book: $\Vert T^2 \Vert =\Vert T \Vert^2$ if $T$ is normal?

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a bounded linear operator, and let $T^*$ denote the Hilbert adjoint operator of $T$. I can show that if $T$ is normal (i.e. $T T^* = T^* T$), ...
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Why does s = z+1?

What exactly is Laplace transform? motivated me to ask why unit function is $1/s$ by Laplace transform and $1/(1-z)$ by Z-transform? Both seem to be integrals of delta-pulse and secondary integration ...
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24 views

What are these algebraic properties called?

Suppose $O$ is some operator, suppose $f$ and $g$ are both functions, then linearity implies that: $O(\alpha f + g) = \alpha O(f) + O(g)$ What about the following property: $(O_1+O_2)(f) = O_1(f) ...
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21 views

The spectral projections of convolution operator

Given a self-adjoint operator $A$ in a Hilbert space $H$. How can one find its spectral projections $\{E_{\lambda}\}_{\lambda\in\sigma(A)}$? In particular, given a convolution operator on $L^2(G)$, ...
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33 views

Is the Cesaro Operator normal?

The Cesàro operator $T:ℓ_p→ℓ_p$ is defined by $$(Tx)_k=(1/k)\sum_{j=1}^k x_j$$ where $x=(x_j)$. Is this operator normal?
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Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
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is $T$ compact operator?

is $T$ compact operator? $T:C[0,1]\rightarrow C[0,1]$: $x(t)\mapsto x(t^2)$ where $t\in[0,1]$ with supremum norm Could you please help.
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50 views

Eigenvalues and eigenvectors of certain diagonal constant matrices

Suppose I have an infinite complex diagonal constant (Toeplitz) matrix, which is also Hermitian. This is given by finite number of complex parameters $z_1, z_2, \cdots, z_k$. If, $z_1$ is the ...
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74 views

Solve the equations $\|Av\|=1/\|A^{-1}w\|$, $\|w\|=1$

I'm sorry if my question is rather stupid, but I have a brainfreeze right now. I want to prove that, for every $A\in GL(2,\mathbb{R})$ and for every $v\in \mathbb{R}^2$, $\|v\|=1$, I can find $w\in ...
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Prob. 8, Sec. 3.10 in Kreyszig's functional analysis book: An isometric linear operator has its adjoint as its left inverse

Let $H$ be a Hilbert space, and let $T \colon H \to H$ satisfy $$\langle Tx, Tx \rangle = \langle x, x \rangle \ \mbox{ for all } \ x \in H.$$ Then $T$ is bounded and norm $\Vert T \Vert = 1$ (unless ...
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20 views

Prob. 6, Sec. 3.10 in Kreyszig's functional analysis book: Powers of self-adjoint operators

Let $H$ be a Hilbert space. If $T \colon H \to H$ is a bounded self-adjoint linear operator and $T \neq 0$, then $T^n \neq 0$ for all $n \in \mathbb{N}$. How to show this? I've managed to show ...
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49 views

generalizations of continuous operators?

What are generalizations of the notion of continous linear operator $P:X\to X$, where X is a Banach space? I'm looking for some broader class of operators that nevertheless share some properties of ...
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44 views

Norm of the quotient map for a normed space [duplicate]

Let $X$ be a normed space and $F$ a closed subspace. On $X/F$ let us take the quotient norm $||[x]|| = \inf_{y \in F} ||x - y||$. Consider the quotient $q : X \rightarrow X/F$. I can see that, if ...
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12 views

Bounded linear operator can be a locally essentially bounded function?

We know that, a bounded linear operator, is a linear transformation $L:H\rightarrow H$ on Hilbert space $H$ such that $$\|Lv\| \le M \|v\|, \ \ \ (M>0, v\in H)$$ A bounded linear operator is ...
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9 views

a inequality about norm of operator with hormander's condition

Let the system of vector fields $X=(X_{1},X_{2},\cdots,X_{m})$ satisfies the Hormander's condition on $\Omega$ with Hormander indes is $Q\geq 1$.Will there exist some estimate like ...
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1answer
26 views

Frame operator on finite dimensional Hilbert space.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$: ...
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25 views

Bitwise Operations and the Naming Convention of their Operators

So I just recently came across a bitwise operation on StackOverflow which shifts the bits in a binary number to the right while zero-filling from the left. The left side zero-filling overwrites the ...
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1answer
13 views

Associated matrix to operator on infinite dimensional spaces.

A linear operator on a vector space has a basis through which write its associated matrix. This is certainly true for finite dimensional spaces. But is it still true for infinite dimensional spaces? I ...
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22 views

A question on bounded operator $\|T_A x\|\leq K\|x\|$

We consider an operator $T_A:H\rightarrow H$, where $H$ is an Hilbert Space and $A$ is its associated $N\times N$ matrix. $T_A$ is said "bounded" if there exists a constant $K>0$ such that $$\|T_A ...
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Prove that matrix $[S]$ associated to operator is such that $A |\zeta|^2\leq s_{ij}(x) \zeta_i \zeta_j\leq B |\zeta|^2$.

Let us consider $N\times N$ matrix $[S]$ associated to operator $S:V\rightarrow V$ where $V$ is a Hilbert space; $S$ is linear, bounded, invertible, positive and self-adjoint. Prove that $[S]$ is ...
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0answers
41 views

Spectral theorem for compact normal operators

Let $H$ be a Hilbert space and $A$ a compact normal operator from $H$ to $H$. How to show that its eigenspaces produce the space? I can show it for self-adjoint operators and by setting ...
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2answers
49 views

How to find the poles of a green function?

I am trying to construct a green function for $y''+\alpha^2u=f(x), u(0)=u(1), u'(0)=u'(1)$. For that I am trying to follow the procedure described here:(Construct the Green s function for the ...
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85 views

Schrodinger Operator with Finite Discrete Spectrum in $(-\infty, -1]$

I'm reading parts of Reed and Simon's Analysis of Operators and have come across a statement I find puzzling. They say that if $V$ is a bounded function of compact support on $\mathbb{R}^3$ then ...
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23 views

Showing that an operator is bijective

Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain ...
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36 views

$\overline{\mathrm{Im} (T^*T)} = \overline{\mathrm{Im} T^*}$

I need to prove that in a Hilbert space, $\overline{\mathrm{Im}(T^*T)} = \overline{\mathrm{Im}T^*}$. I have already shown that $\ker (T^*) = (\mathrm{Im} T)^\perp$ and have so far concluded that ...
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41 views

An invertible sparse matrix?

I'm not entirely certain about how to tackle this problem.... I hope you ladies and gents can help :) If $M\in M_{n\times n}(\mathbb{R})$ be such that every row has precisely tow non-zero entries, ...
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1answer
94 views

Showing that an operator generates a contraction semigroup

Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain ...
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228 views

Proving that $\text{ri rge}\,A=\text{ri conv rge}\,A$

"If $A:\mathbb R^n\rightrightarrows\mathbb R^n$ is maximal monotone,then $\text{ri rge}\,A$ is convex". This is a proposition in auslender's book about the asymptotic cones. We can prove that ...
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1answer
22 views

Connection between Stinespring's factorization theorem and the spectral theorem for bounded operators

I know at least 2 versions of a Spectral theorem for operators, one of them is the following Theorem: Let H be a separable complex Hilbert space, $A\in L(H)$ self-adjoint ($L(H)$ are the bounded ...
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19 views

Origins of the Cesaro Operator

I am wondering when the Cesaro Operator was first studied. I can find an article from 1965 but I'm wondering if there are any previous ones.
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2answers
36 views

Inverse of $I +T^*T$

I am trying to show that the inverse of the operator $I +T^*T$ exists. What I have been trying to do is trial and error taking inverses of $T$ and $T^*$ but to no avail.
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3answers
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$A$ and $B$ are bounded linear operators from the normed linear space $X$ to itself. If $AB$ is invertible are $A$ and $B$ invertible?

I think I understand the proof for square matrices, such that $(AB)^{-1}=B^{-1}A^{-1}$, but I'm not sure if I can just say the same for the bounded linear operators A and B. Does the existence of ...
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1answer
7k views

Magnitude of a Matrix?

Consider a vector V. The magnitude of this vector (if it describes a position in euclidean space) = distance from the origin is simply: $(V^TV)^{1/2} $ aka the square root of the dot product... ...
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Why is the Calkin algebra purely infinite?

I tried using the fact that in a simple unital $C^*$-algebra, $\mathcal{A}$, purely infinite is equivalent to the following: If $x\in\mathcal{A}$ is non-zero, then there exists $a,b\in\mathcal{A}$ ...
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Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
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45 views

Prove that norm of Yosida approximations blows up

I need an help with the following exercise about Yosida approximations. Let's fix the notation: let $A$ be an unbounded linear operator define on a dense domain $D(A)\subset X$ of a Banach space $X$. ...
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1answer
30 views

Weak operator limit of projections in $B(H)$

Let $H$ be infinite dimensional and $\cal P$ be the set of all projections in $B(H)$. Show that $\cal P$ is weak operator dense in $(B(H))^+_{\|.\|\leq 1}$, the set of positive operators in the unit ...
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1answer
14 views

Show that an operator is closable

Let $H=\mathcal{L}^2(\mathbb R^2,dxdy)$ and let $A$ the operator defined by: $$ A[f](x,y)=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}+i(y\frac{\partial f}{\partial ...
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2answers
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Does there exists an operator with these properties?

Consider with $(\Omega,\Sigma,\mu)$ a $\sigma$-additive measure space. Is there a linear operator $P \neq 0$ $$P : L^1(\mu) \to L^1(\mu) $$ which fulfills $$ \|Pf \| \leq \|f\|,$$ $$ f\geq0 ...
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0answers
43 views

Show $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$

Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = ...
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Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
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270 views

Where does the notation $\mathrm{Ad}(U)$ for $a\mapsto UaU^*$ come from?

I have often seen, in the context of operator theory and operator algebras, the notation $\mathrm{Ad}(U)a=UaU^*$, where $U$ is a unitary operator on a Hilbert space $H$ and $a$ is a bounded linear ...
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42 views

Derivation of perturbation series

I'm a little bit confused about the derivation of the perturbation series. I know from my quantum mechanics course that for a perturbed operator, eigenvalue is a series that is depend on the ...
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1answer
42 views

When can we exchange the trace and an integral/limit/derivative?

For a trace class operator $A$ (acting on a Hilbert space), that is parameterised by a real variable $x$, what are the conditions for the following to hold? $$ \mathrm{tr} \int_a^b A(x) \, dx = ...
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1answer
28 views

Is it necessary to use the Hahn-Banach theorem to show that $(X/M)^*\simeq M^\perp$?

Let $X$ be a Banach space with dual space $X^*$, and let $M$ be a closed subspace of $X$. Then $M^\perp=\{x^*\in X^*: x*(m)=0 \text{ for all } m\in M\}$ is a closed subspace in $X^*$. I read the ...
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3answers
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looking for help with a trace/norm inequality

I'm trying to understand a derivation that seems to claim that $\left\vert\text{Tr}\left[\rho U^\dagger\left[U,O\right]\right]\right\vert\leq\|\left[U,O\right]\|$, where $\rho$ is Hermitian and has ...
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1answer
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symetric closed operator and extension [closed]

i have this question let A a symetric closed operator let pose that A have a self adjoint extension is possible that A has an extension such that closure A can't have a self adjoint extension
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1answer
373 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
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1answer
22 views

What can one assume about $T^*$ when showing that $T$ is normal?

Consider a continuous and linear operator $T$ such that $$ T : l^2 \to l^2 $$ where $(a_n) \mapsto (\alpha_n a_n)$ Moreover $(\alpha_n)$ is a sequence of complex numbers that converges to zero. Now, ...
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1answer
80 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ ...