# 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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### Spectral properties of operator over $L^2(0,1)$

Let be $I=[0,1]$ the unit interval and define the operator $$(Af)(x)=\int_0^x f(t)dt$$ with domain $C_0^{\infty}(I)\subseteq L^2(I)$. I want to show the following: $A$ is bounded and compact; ...
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### Is intersection of a dense subspace and a closed subspace of a Hilbert space also Dense?

I have a Hilbert space $H$ and a closed operator $T$ defined on its domain $D(T)$ which is dense in H. Also $M = \text{range} \ T^n$, for some $n$, is given to be closed. Consider the restriction of ...
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### Let $T$ be a definite integral operator on $(C[a,b])$. Find function $k_j$ such that $T^j (x)=\int_{a}^{t} k_j (s,t) x(s) \,ds$

This is the last part of chain of related questions: I was asked to prove that $$T:C([a,b]) \to C([a,b])$$ given by $$Tx(t)=\int_{a}^{t} x(s) \, ds$$ is linear bounded operator on ...
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### linear function, operator norm

Let be $\Phi:V\to W$ a linear function between the vector spaces $V$ and $W$ with the norms $\|\cdot\|_V$ and $\|\cdot\|_W$. Prove that $$\|\Phi\|_{\mathcal{L}(V,W)}=\kappa_{abs},$$ while ...
The Associated Legendre operator is $$L_mf = -\frac{d}{dx}\left((1-x^{2})\frac{df}{dx}\right)+\frac{m^{2}}{1-x^{2}}f,$$ where $m$ is a positive integer. For the purposes here, define ...