Questions tagged [compact-operators]
A compact operator is an operator from normed space $X$ to a normed space $Y$, such that image of every bounded subset of $X$ is relatively compact in $Y$. It's used with (functional-analysis) and (operator-theory) tags.
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Spectrum of $S_r$ and $S_l$ on $\ell _2 (\mathbb{Z})$
Calculate the spectrum of $S_r$ and $S_l$ on $\ell _2 (\mathbb{Z})$
I want to make sure my solution is correct:
$\sigma _r (S_r)$: we need $S_r (a_n) = \lambda (a_n)$ and so we get for every $n=: a_{...
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Compact + absolute convergence of eigenvalues $\Rightarrow$ trace class?
Let $H$ be a complex Hilbert space. Lidskii’s theorem says that if an operator $T \in B(H)$ is trace class, then $\operatorname{tr}(T) = \sum_i \lambda_i$, where the sum includes all nonzero ...
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Is the set of Finite rank Operator closed compared to compact operators set
Let K(X,Y) be the set of compact operators from X to Y. Let F(X,Y) be the set of finite rank operators from X to Y. Is F closed sub space of K?
I proved every finite rank operator is compact. But I'm ...
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$T$ compact operator, Let $\Delta^*_{\bar{\lambda}}$ subspace of $X^*$. Prove $\Delta^*_{\bar{\lambda}} = \bar{\Delta^*_{\bar{\lambda}}}$
We have proved the following claim:
Let T be compact operator and $\lambda \neq 0$. Then $\Delta_\lambda = \bar{\Delta _\lambda}$.
Now there is the corollary:
$T$ compact operator, Let $\Delta^*_{\...
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Is the Riemann Liouville fractional integral compact operator? [closed]
I am about to figured out that is the Riemann Liouville fractional integral compact operator or not? where f is continuous function in [0, b].
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Perturbation of semi-Fredholm operators in Frèchet spaces
It is well known that the index is a continuous function on the set of Semi-Fredholm operators on a Banach space, and even on quasi-Banach spaces.
The result is unfortunately false in general Fréchet ...
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Divergent Tail Sums of Approximations of Non-trace Class Compact Operators
I'm working on approximations of compact operators that are not trace class, and I'm looking for ways to provide meaningful approximation error estimates for truncated eigenfunction expansions. I ...
user1328882
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Convolution preserve the boundary condition
Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function on some interval $[-L,L]$, and $u$ is some 'good enouth' function on $[0,L]$ that ...
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Compactness of an operator on l^2
I am working on some problems from Kreyzig's Functional Analysis and got stuck on this problem:
Let $T: l^2 \rightarrow l^2$ be defined by $Tx = y = (\eta_j)$ where $x = (\xi_j) $ and $\eta_j = \sum_{...
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Is the notion of compactness of operators preserved under quasi-similarity?
Let $H$ be a Hilbert space and $T \in \mathcal L (H).$ Then $T$ is said to be quasi-invertible if it is one-one and has dense range. The operator $T$ is said to be quasi-similar to an operator $S \in \...
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What is a conjugate unitary operator?
I'm studying operator theory and I have encountered the concept of a cojugate unitary operagtor several times. However, I cannot find any reliable references. There is one paper which claims that a ...
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$u^*$ comapact implies $u$ compact proof verification
Suppose that $X, Y$ are Banach spaces and suppose that $u \in L(X, Y)$, with $u^*$ compact, where $u^*: Y^* \to X^*$, given by $u^*(\tau) = \tau \circ u$ is the adjoint of $u$. I am asked to prove ...
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An interesting family of seminorms $\mathcal F$ and comparison between the topology generated by this seminorms and the Weak Operator Topology.
I am learning functional analysis and I am stuck with the following questions from Strong Operator Topology and Weak Operator topology on $\mathcal B(H)=\{T:H\to H:T$ is Op-Norm continuous,linear $\}$....
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Proving that operator in $L^2[0,1]$ is compact
I need help with some functional analysis:
Let $A$ be a continuous linear operator on $L^2[0,1]$ and for any $f \in L^2[0,1]$ the function $Af$ is Lipschitz continuous. Show that $A$ is compact.
It is ...
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I am trying to show that $K: H_0^1(\Omega) \rightarrow H_0^1(\Omega)$ is a compact operator.
$\textbf{Suppose.}$
$\Omega$: bounded open in $\mathbb{R}^n$,
${W^{1,2}}(\Omega)$: the space of $u:\Omega \to\mathbb{R}$ in $L^2(\Omega)$ with a weak derivative $Du$ in $L^2(\Omega)$, and
${H_0}^1(\...
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