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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Dual norm of the matrix L1 norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
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102 views

When is an operator on $\ell_1$ the dual of an operator on $c_0$?

Suppose $T:\ell_1\to\ell_1$ is a continuous linear operator. When can we say that $T$ is a dual, or adjoint, of an operator on $c_0$? In other words, under what conditions can we find a continuous ...
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30 views

Fixed point of projected operator

Let $X \subset \mathbb{R}^n$ be a compact convex set and let $f: X \rightarrow X$ be Lipschitz continuous and such that $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \leq 0 $$ for all $x, y ...
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22 views

Commutant of a set of operators and norm topology.

In the references I have it's remarked that the commutant $S'$ of a set $S$ in $B(H)$, where $H$ is a Hilbert space, is closed in the weak operator topology. And this is true because if ...
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1answer
47 views

Linear operator in $\ell^2$

Let $A \colon \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ be the linear operator defined by $\left( Ax \right)_k = \sum_{i \in \mathbb{Z}}a_{ki}x_i$, where $a_{ki} = 1/(k-i)^2$ if $k \neq i$ and ...
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1answer
29 views

Proof of operator norm with the equality of involution

Given the equality $$\|A^*A\| = \sup_{\| x \| = \| y \| = 1} | (Ax, (A^*)^*y) | $$ How do show that it is equal to $\|A\|^2$ Is it by using Cauchy Schwartz inequality such that $(x, y) \leq ...
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26 views

Continuous extension of differential operator to sobolev spaces

If $T$ is an differential operator of order $k$ from $\mathbb{C}$-vector bundle $E$ to $F$ over a compact differential manifold $X$. Question: how can we extend it to a continuous linear map between ...
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69 views

Existence of a semigroup of bounded operators which is not $C_0$

Let $X$ be any Banach space. Then we can define a $C_0 $ semi group of bounded operators on $X$. But my question is that can we define a semi group of bounded operators which is not $C_0$?
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Boundedness of Volterra operator with Sobolev norm

Consider the subspace of $C^\infty([0,1])$ functions in the Sobolev space $H^1$. I want to know whether the Volterra operator \begin{equation} V(f)(t) = \int_0^t f(s) \, ds \end{equation} is bounded ...
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57 views

$n$th derivative of $f(x)$ using limit definition

After playing around with the limit definition of the derivative for higher order derivatives, I noticed the following odd relationship to determine it for an nth order derivative: Let $F^n=f(x+nh)$ ...
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17 views

Duality relation with respect to differential operators

I have the following differential operator $$L^{\pm}h(x,y)=\pm\frac{x}{2}\frac{\partial h}{\partial x}(x,y)\pm\frac{y}{2}\frac{\partial h}{\partial y}(x,y)+\frac{1}{2}\frac{\partial^2 h}{\partial ...
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15 views

Coordinate Change Operator

Let $ f: \mathbb{R} \rightarrow \mathbb{R} $ be analytic. Recall that for $ h \in \mathbb{R} $, the translated function $ \tilde{f} (x) = f(x+h) $ can be formally written as $ \tilde{f} = e^{ h ...
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42 views

Show there exists a unique solution to $-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$

Let $\lambda\in (-1,1)$. Show that for every $f\in C[0,1]$ there exists a unique solution $u\in C[0,1]$ to $$-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$$ With $u(0)=u'(1)=0$. My work thus far: ...
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1answer
32 views

Kernel of the Extension of a Bounded Linear Operator

Suppose $T\colon E\to F$ is a bounded linear operator between Banach spaces. Moreover let $i\colon E\to E’$ be a dense, compact inclusion of $E$ into some other Banach space $E’$. Finally assume that ...
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86 views

Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy $

Consider the operator $$ Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy $$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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102 views

What is the difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...
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1answer
64 views

Condition on the kernel of the integral operator to belong to the trace class?

Let $\mu$ be a finite compactly supported Borel measure on the real line. Consider the integral operator $K$ on $L^2(\mu)$, $$ (Kh)(x)=\int h(y)k(x-y)\, d\mu(y), $$ where $k$ is a fixed function. ...
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64 views

Proving unitary inequivalence

Concerning a finite dimensional vector space, we can determine whether two matrices are unitarily equivalent or not by depending on some similarity invariants or Specht's theorem. If we consider an ...
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41 views

Prove that for a bounded self adjoint operator, $\langle Tx,x\rangle \geq 0$ is equivalent to $\sigma(T)\subset [0,\infty)$

Prove that for a bounded self adjoint operator, the following are equivalent: A: $\langle Tx,x\rangle \geq 0$ B: $\sigma(T)\subset [0,\infty)$ What I have said so far: Since $T$ is self adjoint, ...
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1answer
35 views

$C^{*}$ algebras positively dominated by finite dimensional algebras

Assume that $A$ is a $C^{*}$ algebra and $B$ and $C$ are two sub $C^{*}$ algebras of $A$ such that: $B$ is finite dimensional algebra. For all positive $c\in C$, there exist a positive $b\in ...
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1answer
50 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
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66 views

For a Hilbert space $\mathcal{H}$, is every bounded linear operator on $\mathcal{H}$ a linear combination of unitary operators?

Let $(\mathcal{H}, (\cdot, \cdot))$ be a Hilbert space, and let $B \in \mathcal{B}(H)$ be a bounded linear operator on $H$. If $\mathcal{H}$ is a complex Hilbert space, then $B$ can be written as a ...
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117 views

How to show that $e^{tA}=\frac{1}{2\pi i}\int_{\{Re \ \lambda =a\}}e^{\lambda t}(\lambda I-A)^{-1}d\lambda$?

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We can show that if $|\lambda|>|A|$ then $\lambda I-A$ is invertible and $$(\lambda ...
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1answer
25 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
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27 views

Bounding the operator norm of some operators

Let $T:X \to Y$ be an operator between Banach spaces with $Tx:= A(Bx+Cx)$ where $A$, $B$ and $C$ are operators too. Is it possible to find a upper bound of the form $$\lVert{(I+T)^{-1}} \rVert \leq ...
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47 views

Sufficient condition for an operator to be compact in Hilbert space of holomorphic function with respect to Gaussion weight (Fock space).

What I read in a book I could not understand, some one please help. Let $\mathcal{F}=\{f:\mathbb{C^n}\rightarrow\mathbb{C}: \text{$f$ is holomorphic and}\int_{\mathbb{C}^n}\lvert ...
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1answer
58 views

Convergence of square root operators

Let $Q_n$ and $Q$ be compact positive and symmetric operators. Let $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. Given $Q_n$ converges to $Q$ w.r.t. operator norm. Does $A_n$ converges to $A$? Thanks. ...
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A possible Corollary of the Fredholm alternative?

Let $H$ be a Hilbert space, $P : H \rightarrow H$ a positive-definite (bounded) operator and $K : H \rightarrow H$ a compact (not necessarily self-adjoint) operator. Let $T = P + K$. In particular, ...
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1answer
21 views

Definitions of hemicontinuity

can anyone see the equivalence or relation between the following two definitions of hemicontinuity that I encountered: Assume that $K$ is a closed, convex subset of Banach space $X$. Let $X^{*}$ be ...
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1answer
31 views

Skew-adjoint differential operator $B$ with spectrum $\sigma(B)=i(-\infty,-1]$

Consider the Hilbert space $X=L^{2}\left(\mathbb{R}^n\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\mathbb{R}^n)$. It is known that the spectrum of $A$ is ...
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1answer
20 views

Problem involving pseudomonotone mappings on Banach space

I have the following question regarding mappings on a Banach space $X$. If anyone has an idea or hint as to how to resolve this question it would appreciated. Let $X$ be a Banach space, $X^{*}$ its ...
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78 views

How to express $e^{yS^2}(f(x))$ in closed form where $\frac{d}{dx}=S$

$$ f(x)+\frac{y.f'(x)}{1!}+\frac{y^2 f^{''}(x)}{2!}+\cdots=e^{yS}(f(x))=f(x+y) \text{ where }\frac{d}{dx}=S$$ is a operator $$ f(x)+\frac{y.f''(x)}{1!}+\frac{y^2 ...
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1answer
35 views

$R\mbox{ is a right multiplier and }R(a)b=a\overset{?}{\implies} A\mbox{ is unital }$

Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that $$ \exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad $$ implies $A$ is unital. I know this is true if A is a ...
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16 views

T=T** for densely defined operator T

Suppose that $H_1$ and $H_2$ are Hilbert spaces and $T: H_1 \to H_2$ is a densely defined linear map with closed graph. Show that $T = T^{**}$. (I have shown that such a $T$ has a densely defined ...
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52 views

Spectra of operators on different spaces

Can the same operator when defined on two different spaces have different spectra? For example and operator defined on $C_0$ and on $\ell_2$?
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46 views

Spectrum of symmetric, non-selfadjoint operator on Hilbert space

I heard that any (unbounded) densely defined and symmetric operator $A: \text{dom}(A)\subset \mathcal{H} \to \mathcal{H}$, which is not selfadjoint, has $\text{spec}( A )= \mathbb{C}$. $\mathcal{H}$ ...
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1answer
58 views

Spectrum of a finite rank operator

If $ T\in B(H)$ is a finite rank operator, then there are orthonormal vectors $e_1,...,e_n$ and vectors $g_1,...,g_n$ such that $Th=\sum_{i=1}^n (h,e_i )g_i$, then we can easily see that $T$ is ...
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1answer
48 views

Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$ where $T_k(f):=\frac{1}{2} ...
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When does an integral operator belong to the Schatten - von Neumann class in terms of its kernel?

It is well known that an integral operator $X: L^2(\mu)\to L^2(\nu)$ with kernel $k(x, y)$ belongs to the Schatten -- von Neumann class $\mathfrak S_2$ if and only if $\int |k(x, y)|^2\, d\mu(x)\, ...
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1answer
38 views

What is the domain of an operator?

There seems to be a lot confusion on this notion of a domain of an operator $D(A)$ where $A$ is an operator. Can someone use a simple example to illustrate exactly what this is? Say, let $A$ be a ...
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47 views

Prove that operator is completely continuous

Let's consider Banach space $\ell^\infty$ of bounded sequences $x = \{ \xi_n\}_{n=1}^\infty$: $$ ||x|| = \sup_{n \in \mathbb N} |\xi_n|. $$ Suppose matrix $||a_{i j}||_1^\infty$ specifies operator $A$ ...
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1answer
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Let $A: X \to X$ be a Fredholm operator, then $Ax=y$ has a solution iff $Ax=0$ implies $x=0$?

Let $X$ be a Banach space and let $A: X \to X$ be a Fredholm operator, then $Ax=y$ has a solution iff ($Ax=0$ implies $x=0$)? I can't see how this is implied by the common definitions of Fredholm ...
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1answer
43 views

Relation between $\epsilon$-pseudospectrum of operators

If $H$ is a Hilbert space and $\sigma_{\epsilon}(T)$ denotes the space of all $\epsilon$-pseudospectrum of the operator $T$ and $S, T\in B(H)$ be such that $TS=ST=0$, why ...
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44 views

local inverse of polynomial

Is there a possibility to invert a polynomial locally? I've got the following problem, concerning control theory: Imagine a ideal amplifier with a feedback loop: Let firstly A be not dependent on ...
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1answer
20 views

Solution sets/ existence and uniqueness of solutions to $Ku-\lambda u=\int^1_0 \frac{x^2}{1+y^3}u(y)dy-\lambda u(x)=f(x)$

Given $$ Ku-\lambda u=\int^1_0 \frac{x^2}{1+y^3}u(y)dy-\lambda u(x)=f(x) $$ A) For what values of $\lambda$ does there exist a unique solution for all $f\in L^2(0,1)$? B) Find the solution set ...
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66 views

Generalized Leibniz Rule

Leibniz Rule states that, $$(f\cdot g)^{(m)}(x)=\sum_{k=0}^m \binom{m}{k} f^{(m-k)}(x)g^{(k)}(x).$$ Writing this with differentiation denoted by $D$, we might say $$D^m (fg) = \sum_{k=0}^m ...
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3answers
64 views

Prove that if $T=T^*$ and $\sigma(T)=\{\lambda\}$, then $T=\lambda I$

Show that if $T$ is a self adjoint linear operator on a Hilbert space such that the spectrum contains a single point $\lambda$, then $T=\lambda I$. Then, show this is false if $T$ is not self ...
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2answers
114 views

Ways to calculate the spectrum of an operator

Friends, I am learning some very basic stuff of spectral theory and kind of lost, in some sense. I am trying to find ways to compute the spectra of different operators, when they work and don't work. ...
2
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1answer
55 views

Maximal subspace on which an operator is bounded

Consider the Banach space $X=C[0,1]$ of real continuous function on $[0,1]$ equipped with the supremum norm. Consider the operator $A:D(A)\to X$, $Af=f'$ for each $f\in D(A)=C^1[0,1]$. We can see that ...
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20 views

Are all cyclic representations irreducible?

I know that for a representation $\pi$ of a $^*$-algebra $\mathcal{A}$ on a Hilbert Space $\mathcal{H}$, if $\pi$ is irreducible then it is cyclic. Is the reverse implication also valid - i.e. is ...