# Tagged Questions

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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### What is a biorthogonal system?

What does biorthogonal mean ? If they say let the system $l^1,l^2,l^3,...,l^n$ Be biorthogonal to the bases $x_1,x_2,x_3,...,x_n$ Of the kernel of $Λ$ So that $Λ$ Is a linear operator
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### Finding the spectral decomposition of $\Delta= \frac{d^2}{dx^2}$ [closed]

What is the spectral decomposition of the operator $\Delta= \frac{d^2}{dx^2}$ in $(L^{2}(\mathbb R), dx)$? Thanks you in advance
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### Spectrum and resolvent of an operator

So for the operator $A:l_2(\Bbb C)\to l_2(\Bbb C)$ defined as: $$A(x_1,x_2,\cdots,x_m,x_{m+1},x_{m+2},\cdots) = (x_1,x_2,\cdots,x_m,0,0,\cdots)$$ We can find the adjoint operator $A^*$ by looking at:...
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### Weak convergence = norm convergence for trace class operators?

Given a (separable) Hilbertspace $H$, I look at the traceclass operators $\mathfrak{S}_1$. I recall the fact that the weak convergence implies norm convergence in the sequence space $\mathcal{l}^1$. ...
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### Show that the following operator (on a Hilbert space) is continuous.

"Let $\mathcal H$ be a complex Hilbert space and let $y\in\mathcal H.$ Show that the linear transformation $f:\mathcal H\to\mathbb C$ defined by, $f(x)=\langle x,y\rangle$ is continuous." Here ...
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### Approximation property for Banach space and $l^{p}$

Let's consider a compact operator $T: X \rightarrow l^{p}, 1 \leq p < \infty$. I would like to check, whether it's possible to approximate $T$ by the operators of a finite rank with respect to an ...
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### Show that the following operator is not a surjection.

"Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Show that $T$ is not a surjection". Here is what ...
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### Find the norm of the following operator.

Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Find $\|T\|$. I was hoping to solve this problem ...
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### A system of equations

Let $H$ be a non-separable Hilbert space. Assume $E$ is an orthonormal basis in $H$. Let $E_0=\{e_n\}$ be a countable subset of $E$ and let $\{\zeta_n\}$ be a bounded sequence in $H$. Let $E_1$ be a ...
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### Norm of the inverse of a map $\ell^2\to\ell^2$

Let $Au_i=u_{i+1}-(2-\beta)u_i+u_{i-1}$ whith $u\in \ell^2=\{(u_i)_{i\in \mathbb Z}, u_i\in \mathbb R:\sum_{i\in \mathbb Z}u^2_i<+\infty\}; \beta>0$. How to compute $||A^{-1}||$ or estimate it? ...
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### Is the set of linear combinations dense in the set of the dual space of $l_p$?

Good day, Right now I'm working with the book "Functional Analysis" by Bachman and Narici, it is available on Google Books, see https://books.google.de/books?id=wCHtLumoGY4C&printsec=frontcover&...
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### Fréchet differentiability of Nemyckij operator defined on $L^2$

I have been told the following. Suppose $\Omega\subseteq\mathbb{R}^n$ is a bounded borel set, $f$ is Carathéodory function on $\Omega\times\mathbb{R}=\{(x,s):x\in\Omega,s\in\mathbb{R}\}$, $f_s$, ...
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### Showing that the operator is bounded and find its norm.

I have this operator $T: L^p(0,\infty)\rightarrow L^p(0,\infty)$, $1<p<\infty$ : $(Tf)(x)=1/x\int_0^xf(t)dt$. I am supposed to show that it is bounded and fint its norm. I had an idea that ...
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### $A^2=A$ prove $A$ is hermite matrix

Let matrix $A\in M_n(\mathbb{C})$ satisfying $A^2=A$ , for every $n\times 1$ vector $x$ we have $|Ax|\le|x|$ where $| |$ denotes the usual norm of vector , prove $A$ is a hermite matrix.
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### Derivation of an integral equation

I have the following system $$\frac{d}{dx}\left(a(x)\frac{du}{dx}\right)=f, \text{ for } x \in (0,1)$$ with boundary conditions $u_x(0)=0$ and $u(1)=0$. For $a(x)>0$, and $b(x)=\frac{1}{a(x)}$, I ...
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### An operator satisfying in a sequence of equations

Assume that $H$ is a non-separable Hilbert space. Let $\{\eta_n\}$ be an arbitrary sequence in $H$. Let $\{\zeta_n\}$ be a sequence in $H$ which forms a linearly independent set. Does there exist ...
### A translation invariant sigma algebra in $B(H)$
Assume that $H$ is a non-separable Hilbert space. Let $s_0$ be the family of all basic neighborhoods in the strong operator topology. We denote $M_s$ by the sigma algebra generated by $s_0$. ...