# Tagged Questions

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### Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
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### equivalence of statements involving compact operators

Let T be a compact operator on a hilbert space H.I want to show that the following 2 statements are equivalent. For each $\lambda \in {\bf C}$, let $V_\lambda = \{ x \in H: Tx = \lambda x \}.$ ...
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### Show there exists a unique solution to $-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$

Let $\lambda\in (-1,1)$. Show that for every $f\in C[0,1]$ there exists a unique solution $u\in C[0,1]$ to $$-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$$ With $u(0)=u'(1)=0$. My work thus far: ...
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### Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy$

Consider the operator $$Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy$$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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### Prove that for a bounded self adjoint operator, $\langle Tx,x\rangle \geq 0$ is equivalent to $\sigma(T)\subset [0,\infty)$

Prove that for a bounded self adjoint operator, the following are equivalent: A: $\langle Tx,x\rangle \geq 0$ B: $\sigma(T)\subset [0,\infty)$ What I have said so far: Since $T$ is self adjoint, ...
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### Why does s = z+1?

What exactly is Laplace transform? motivated me to ask why unit function is 1/s by Laplace transform and 1/(1-z) by Z-transform? Both seem to be integrals of delta-pulse and secondary integration ...
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So here is my question, Let $H$ be a Hilbert-space and $K:H\rightarrow H$ a compact operator. I know that if $K$ is self adjoint, then it has one eigenvalue $\mu$ such that $|\mu|=||K||$. Can some ...
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### Doubt on eigenvalues of normal operators

I'm trying to understand the solution of the following problem: $T$ is a normal operator. If $T( v)=\lambda v$, then $T^*(v)=\bar\lambda v$: The solution is: I didn't understand why we ...
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### Common eigenvector of a sequence of compact operators

Let $H$ be a separable, infinite-dimensional Hilbert space and suppose we have a sequence of norm-one compact operators $(A_n)$ on $H$ which all have 1 as an eigenvalue. Can we pass to a subsequence ...
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### Question about the essential spectrum of a negative difinite operator

please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ??? Please help me Thank you.