# 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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### Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open $H:=L^2(\Omega,\mathbb R^d)$ $U$ be a separable $\mathbb R$-Hilbert space $Q:U\to H$ ...
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### $ι:U→V$ is an embedding, $Q:=ιι^*$, $L∈𝓛(ℝ^d)$, $Φ∈\text{HS}(U,ℝ^d)$ $⇒$ $\text{tr}LΦ\sqrt Q(Φ\sqrt Q)^*$ doesn't depend on $ι$

Let$^1$ $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota\in\operatorname{HS}(U,V)$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $u:\mathbb R^d\to\mathbb R$ be twice Fréchet ...
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### Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
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### noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
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### About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
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### continuous and sequentially continuous

If an operator $T: A\rightarrow B$ satisfying for every sequence $\{X_n\}$ weakly converging to $X$, we have $TX_n \rightarrow TX$ in weak topology. Then, is $T$ weak-weak continuous? And in the WOT/...
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### R.Douglas “Banach Algebra Technique Operator Theory” - Chapter 2 issue

Just before 2.37 Corollary (Spectral Mapping Theorem) Douglas says: If $\varphi (z)= \sum_{n=0}^\infty a_nz^n$ is an entire function with complex coefficients and $f$ is an element of the Banach ...
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### For $p(x)\in \Bbb{C}[x]$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $0\le k\le n-1$, show that $p(\lambda)=0\Rightarrow \lambda\in [0,1]$

For a complex polynomial $p(x)\in \Bbb{C}[x]$ of degree $n$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $1\le k\le n-1$, show that $p(\lambda)=0$ means $\lambda\in [0,1]$. I haven't come by any ...
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### Prove or disprove: $\{t^{2k}\}_{k=0}^{\infty}$ complete in $L_2[-1,3]$

Is $\{t^{2k}\}_{k=0}^{\infty}$ not complete in $L_2[-1,3]$?(Here, completeness of a system is equivalent to the density of its span) Obviously many polynomials in the domain will be irreleant, but I ...
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### Finding closure of image of operator

I'm working on an old exam problem: Define for $u \in C^2([-1,1])$ the operator $L$ by $[Lu](x) = - \frac{d}{dx} \left( (1-x^2) u'(x) \right)$. Set $\Omega = \{ Lu \mid u \in C^2([-1,1]) \}$. Find the ...
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### What is the correct definition for positive operator and positive definite operator?

As far as I know those operators are defined as follows: Positive operator is an operator $L: H\rightarrow H$ such that $\langle L\textbf u|\textbf u\rangle \geq0$ for all $\textbf u \in H$ and the ...