Complete normed spaces whose norm comes from an inner product.
6
votes
0answers
217 views
an infinite series expansion in terms of the polylogarithm function
we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though ...
5
votes
0answers
219 views
Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)
As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
5
votes
0answers
180 views
Why is the numerical range of an operator convex?
Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation}
It is a well-known fact that $W(T)$ is a convex subset of the complex ...
4
votes
0answers
110 views
Inverse of Identity plus Volterra operator
consider the following operator or $L_2(0,1)$,
$(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial.
I am trying to construct the inverse of this ...
4
votes
0answers
86 views
What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)?
What is the dual space of $C([0,T];X)$, where $X$ is a Hilbert space? Is it $\operatorname{BV}([0,T]; X^*)$?
As we know, for $C([0,T])$, the dual space is $\operatorname{BV}([0,T])$, but when it is ...
4
votes
0answers
177 views
Spectral theorem for unitary operators
I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
4
votes
0answers
179 views
Sum of operator and adjoint is self-adjoint
In abstract Hodge theory there is the following lemma:
Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
4
votes
0answers
63 views
Relations between spectrum and quadratic forms in the unbounded case
Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
4
votes
0answers
134 views
Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?
I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
4
votes
0answers
132 views
When functions, analytically continued, carry over certain properties
Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
3
votes
0answers
61 views
Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.
For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$
Here is a quick one. If someone could improve this it would be great
Proof
By Cauchy Schwarz, $\langle x,z \rangle ...
3
votes
0answers
192 views
On the weak and strong convergence of an iterative sequence
I have some difficulties in the following problem.
I would like to thank for all kind help and construction.
Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
3
votes
0answers
86 views
Isomorphic Hilbert spaces
As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
3
votes
0answers
45 views
Existence of an ergodic-looking limit in a Hilbert space
This is part of a problem from Reed & Simon's Functional Analysis -- I'll write the problem first.
Let $V$ be a linear transform on the Hilbert space $H$, such that its powers are uniformly ...
3
votes
0answers
314 views
Question about example of non-separable Hilbert space
I have come across the following example of a non-separable Hilbert space:
Why do I need the discrete topology on $I$? Or more generally: why do I need a topology? If we talk about $L^p$ spaces in ...
