1
vote
1answer
37 views

Does there always exist an equivalent geometric problem for a given differential equation?

I recently came up with a method to show for any given differential equation whose solution is the power series there exists an equivalent geometric problem in Hilbert space. I was wondering if ...
4
votes
0answers
97 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
0
votes
1answer
45 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...
1
vote
1answer
17 views

Proof that solution of $\lambda$-affine, linear ODE is entire in $\lambda$

Suppose $F(\lambda)~(\lambda\in\mathbb{C})$ is a linear ordinary differential operator (with, say, domain $D$ dense in some Hilbert space), and is also affine-linear in $\lambda$. Is there a proof ...
0
votes
1answer
22 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
3
votes
1answer
179 views

Equivalent norm in sobolev space H^2

I consider space $H^{2}(0,a)=\{ f\in L^{2}(0,a): f',f''\in L^{2}(0,a) \}$ I define norm $\Vert w \Vert_{H^{2}}:=b\Vert w''\Vert_{L^{2}}$, where b is positive constant. I couldn't proof that it is ...
3
votes
4answers
104 views

reference for strongly continuous semi-groups

At the moment I am trying to understand the proof of the Fredholm property in Salamon's notes on Floer homology. There I came across the notion of an unbounded operator on a (real) Hilbert space which ...
7
votes
1answer
224 views

Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...