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In the following, $T$ is a bounded operator on a Banach space $X$. $T$ is called "power bounded" if $\sup_{n\in \mathbb N}\|T^n\|<\infty$; $T$ is called "mean ergodic" if the Cesàro sums $\frac{1}{n …

Let $X$ be a finite measure space and $D,M$ be bounded linear operators on a $(L^1(X;\mathbb C))^2$. $D$ is a diagonal operator matrix whose entries are multiplication operators by the invertible func …

Consider the Sobolev space $W^{k,p}(\Omega)$ for $k\in \mathbb N$, $p\in [1,\infty]$ and some open domain $\Omega\subset \mathbb R^n$ $^*$. Then it is known that $W^{k,p}(\Omega)$ is an ordered Banach …

Let $X,Y$ be Hilbert spaces, $D$ be a dense subspace of $X$, $T:D\to Y$ be a linear operator, $\tilde{D}:=T(D)$. Assume $T^*T:D\to X$ to be essentially self-adjoint and the generated semigroup $(e^{-t …

Consider a standard integral operator $T$ formally defined by $$ Tf(x):=\int_{K} k(x,y)f(y)dy,\qquad x\in K, $$ where $K$ is a locally compact metric measure space. It is immediate to see that the ope …

I have just discovered this article about heat kernels on graphs. It has been written by a respected theoretical physicist, but seemingly never made it into a peer-reviewed journal. On the other hand, …

Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then $A+B …

A $C_0$-semigroup is a strongly continuous family $\{T(t)\}$ of bounded linear operators on a normed space $X$, indexed in $\mathbb R_+$ and with two additional properties that make it look like an op …

I am reading Integrals of Nonlinear Equations of Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide pr …