Tagged Questions

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What is a predual of the Banach space of compact operators on $\ell^2$?

I am wondering if the space $K(\ell^2)$ of compact operators on $\ell^2$ can have a predual. Thank you in advance for your help.
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Normal Compact Operator: not diagonalizable!

To proposition 5.17 in Weidmann's 'Lineare Operatoren in Hilberträumen' (german version) it is noted that the expansion of compact operators that are normal rather than self adjoint doesn't apply in ...
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Does Hilbert Transform commute with Function Multiplication modulo Compact on $L^p(R)$?

Define Hilbert Transform (HT) as the convolution with the function $1/x$. E. Stein proves in his book Singular Integrals and Differentiability Properties of Functions that HT, when understood as a ...
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Let $(X, ||\cdot||)$ be a real Banach space and $T: X \to X$ a compact operator (so $\{x_n\}_{n=1}^\infty$ bounded implies that $\{Tx_n\}_{n=1}^\infty$ has a convergent subsequence). Let $x_1, \dots, ... 2answers 85 views Totally boundedness of a compact operator [closed] Let$T:\ell_2(\mathbb N)\longrightarrow \ell_2(\mathbb N)$bounded linear operator such that $$T(\{x_n\})=\{x_n/n\}.$$ I need to prove that$TB(\ell_2(\mathbb N))$, that is closed unit ball in ... 1answer 33 views Compact embedding Prove that the embedding$j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$where$\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$and$\|f\|_\infty$denotes the supremum norm, ... 1answer 94 views Direct sum of eigenspaces of a compact operator has finite codimension In an infinite dimensional Hilbert space the orthogonal complement of the (closure) of the direct sum of eigenspaces of a compact normal operator is finite dimensional. Why is this the case? thanks. 1answer 52 views About a compact imbedding of Sobolev spaces I am studying the Compactness lemma ( on page 570) of the article http://projecteuclid.org/euclid.cmp/1103922134. The lemma says (Compactness lemma ): for$0 < \sigma < \frac{2}{N-2}$,$(N ...
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Let $0<\sigma< \frac{2}{N-2}$ with $N \geq 3$. I know that $H^{1}_{\operatorname{rad}}(R^n)$ (radial functions of $H^{1}(R^n)$ ) is compactly embedded in $L^{2 \sigma +2}(R^N)$. Let $(\psi_v )$ ...
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In $X=\ell^p$, $p\in[1,\infty]$ we consider: $$T(x_1,x_2,x_3,\ldots)=(0,x_1,0,x_3,\ldots)$$ Prove that $T$ isn't a compact operator and that $T^2$ is a compact operator. I think I solved the second ...
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Let $H$ be a hilbert space and $T$ a compact self-adjoint operator on it. T is also injective on a dense subspace $U \subset H$ and we also have that $T(H) \subset U$. Now I am asked whether it is ...
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Compact kernel operator on $L^p$ space

Let $\displaystyle U_1 \subset \mathbb R^{n_1}$ and $\displaystyle U_2 \subset \mathbb R^{n_2}$ measurable sets, $\displaystyle 1 < p,q < \infty$ and consider the measurable function ...
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The trace class operators are the dual of the compact operators

I know that the map from the trace class operators $L_1(H)$ to the dual of the compact operators $K'(H)$ given by $A \mapsto tr( \cdot A)$ is an isometric isomorphism. Linearity is obvious by the ...