# 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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### 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 ...
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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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### 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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### Space of compact operators defined on separable Hilbert space

Let $H$ be a separable Hilbert space and $K(H)$ the Banach space of all compact linear maps from $H$ into itself (with the operator norm). Show that $K(H)$ is separable. There is a hint that states ...
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### Linear positive definite bounded operator $T:H\to H$

I was trying to solve this problem: Let $H$ be a Hilbert space and $T:H\to H$ a bounded linear operator such that $$(Tx,x)\geq ||x||^2 \quad \forall x\in H$$ where $(\cdot,\cdot)$ and $||\cdot||$ ...
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### Is $L$ compact if $\| Lf_k \|\rightarrow 0$ for Orthonormal Basis $f_k$?
Suppose $f_k$ is an orthonormal basis of a separable Hilbert space. $T$ is bounded. $\| (T-\lambda_k)f_k \| \rightarrow 0$, $\| (T-\lambda_k)^*f_k \| \rightarrow 0$. $R$ is the diagonal operator with ...