# Tagged Questions

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### Is a Hilbert space $H$ compactly embedded in its dual?

Is a Hilbert space $H$ compactly embedded in its dual? Is it compactly embedded in itself? No idea how to think of this.
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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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### Criteria of compactness of an operator

Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$ we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$. Is it true that $K$ is compact? ...
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### Hilbert space proof

$X$ is a separable Hilbert space and $A\in L(X,X)$ and compact. I need to prove that $A$ is approximately of finite dimension.
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### Show for compact operator $K$, if $||Kf|| < ||f|| \forall f$, then $||K|| < 1$.

I wanted to check my reasoning on proving this statement, and see if anyone had suggestions for other proofs of this fact. Again, the statement is, if $K$ is a compact operator on a Hilbert space ...
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### Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
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### compact and self adjoint square root of an operator

Let H a Hilbert space and $T:H\rightarrow H$ a linear bounded, self-adjoint, positive and compact operator. How can i prove that the square root of T, $\ T^{1/2}:H\rightarrow H$ is also compact and ...
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### Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective.

Given a map $F:X \to X$ where $X$ is a Hilbert space, $F$ satisfying $f(x):=x-F(x)$ is a compact map. $\lim_{\|x\|\to \infty} \frac{(F(x),x)}{\|x\|} = \infty$ I'm seeking to prove that $F$ is ...
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### If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
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### Trace class for operators

Let $\mathcal{H}$ be a Hilbert space and $T: \mathcal{H} \to \mathcal{H}$ a bounded linear operator. The $n$-th singular number ${\mu_{n}}(T)$ of $T$ is defined as the distance from $T$ ...
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### How do I prove that a particular linear operator has an orthonormal basis?

I have to show that if $T$ is a linear operator such that $T: L^2(\mathbb(R)^n) \to L^2(\mathbb(R)^n)$ and $T(f)(x) = \int_{R^n}f(y)g(x,y)dy$, where $g(x,y)$ is an $L^2$ function, that there is an ...
Let $H$ be a separable Hilbert space and let $T:H \rightarrow H$ be a symmetric bound linear map. a) Show that for every orthogonal projection $P$ on $H$ ($P' = P$, $P^2 = P$) PTP is symmetric. b) ...