fa.functional analysis – Compactness of integral operators

We know that if an operator has $L^2$-kernel, then it is Hilbert-Schmidt.
Is there a similar simple criterion to detect compact operators?

In particular, I’d like to know the following: Let $f$ be a Schwartz function on ${mathbb R}^2$ with $mathrm{supp}(f)subset{mathbb R}times J$ for some compact Interval $J$.
Let
$$
k(x,y)=f(e^x,x-y)
$$

Is the operator $T:L^2({mathbb R})to L^2({mathbb R})$, $T(phi)(x)=int_{mathbb R}k(x,y)phi(y),dy$ a compact operator?

fa.functional analysis – How to think action of orthogonal projection operator on Heaviside function of some operator?

Let $mathbb P_l$ stands for the orthogonal projection in $L^{2}(S^{n-1})$ on the space of $mathcal H_{n,l}$ of harmonic homogenous polynomial with degree $l$ in $n$ variable.
By spectral decomposition theorem of Laplacian on sphere we have
$$mathrm{Id}_{L^{2}left(mathbb{S}^{n-1}right)}=sum_{l geq 0} mathbb{P}_{l}, quad-Delta_{mathbb{S}^{n-1}}=sum_{l geq 0} l(l+n-2) mathbb{P}_{l}.$$

By writing Laplacian in polar coordinate we get
$$ r^{2} Delta_{mathbb{R}^{n}}=left(r partial_{r}right)^{2}+(n-2)left(r partial_{r}right)+Delta_{mathbb{S}^{n-1}}$$

Define $Lambda=sum_{k geq 1} k mathbb{P}_{k} $. By orthogonality we have
$$|x|^{2} Delta_{mathbb{R}^{n}}=left(r partial_{r}right)^{2}+(n-2)left(r partial_{r}right)-Lambda(Lambda+n-2).$$

Now take $r=e^t$ we get
$$|x|^{2} Delta_{mathbb{R}^{n}}=partial_{t}^{2}+(n-2) partial_{t}-Lambda(Lambda+n-2)=left(partial_{t}+Lambda+n-2right)left(partial_{t}-Lambdaright)=mathcal{L}$$
We note that for $lambda geq 1$
begin{equation}
mathcal{L}=|x|^{2} Delta_{mathbb{R}^{n}}, quad mathcal{L}_{lambda}=e^{-lambda t} mathcal{L} e^{lambda t}=mathcal{L}_{-, lambda} mathcal{L}_{+, lambda}=mathcal{L}_{+, lambda} mathcal{L}_{-, lambda}
end{equation}

Where begin{equation}
mathcal{L}_{+, lambda}=partial_{t}+lambda+Lambda+n-2, quad mathcal{L}_{-, lambda}=partial_{t}+lambda-Lambda.
end{equation}

Fundamental solution of differential operator $mathcal{L}_{+, lambda} mathcal{L}_{-, lambda}$ is given by
begin{align}
E=& H(lambda-Lambda)(2 Lambda+n-2)^{-1}left(e^{-(lambda-Lambda) t}-e^{-(Lambda+lambda+n-2) t}right) H(t)notag \ &-H(Lambda-lambda)(2 Lambda+n-2)^{-1}left(e^{-(lambda-Lambda) t} H(-t)+e^{-(Lambda+lambda+n-2) t} H(t)right)
end{align}

where $H$ is Heaviside function. i.e. begin{equation}
v=E *left(mathcal{L}_{+, lambda} mathcal{L}_{-, lambda} vright)
end{equation}

Now we wanted to estimate $left|mathbb{P}_{k} v(t)right|_{L^{p^{prime}}left(mathbb{S}^{n-1}right)}$.

For that purpose we write $F=mathcal{L}_{+, lambda} mathcal{L}_{-, lambda} v$.

By property of fundamental solution we have
begin{align*}
v(t)&=frac{H(lambda-Lambda)}{2 Lambda+n-2} int_{-infty}^{t}left(e^{-(lambda-Lambda)(t-s)}-e^{-(lambda+Lambda+n-2)(t-s)}right) F(s) d s\
&quad-frac{H(Lambda-lambda)}{2 Lambda+n-2}left(int_{t}^{+infty} e^{-(Lambda-lambda)(s-t)} F(s) d sright.\
&quadleft.quad+int_{-infty}^{t} e^{-(lambda+Lambda+n-2)(t-s)} F(s) d sright)
end{align*}

Now we apply $mathbb P_k$ to both side.

I can think that
$e^{-(lambda-Lambda)(t-s)}$ can be think as series of exponential in $Lambda$. so after applying $P_k$ to that we get
$e^{-(lambda-k)(t-s)}mathbb P_k$ because of orthogonality.

But I have doubt how to apply $mathbb P_k$ on $frac{H(lambda-Lambda)}{2 Lambda+n-2}$.

I thought about Fourier series of Heaviside function but there is problem about convergence and Heaviside on whole real line is not periodic function as such so I can not get Fourier series. Also how to apply on its denominator $2 Lambda+n-2$. Final expression is given as
begin{align*}
mathbb{P}_{k} v(t)&=frac{H(lambda-k)}{2 k+n-2} int_{-infty}^{t}left(e^{-(lambda-k)(t-s)}-e^{-(lambda+k+n-2)(t-s)}right) mathbb{P}_{k} F(s) d s\
&quad-frac{H(k-lambda)}{2 k+n-2}left(int_{t}^{+infty} e^{-(k-lambda)(s-t)} mathbb{P}_{k} F(s) d sright.\
&quadleft.quad+int_{-infty}^{t} e^{-(lambda+k+n-2)(t-s)} mathbb{P}_{k} F(s) d sright)
end{align*}

Any help or hint will be appreciated.

fa.functional analysis – Is the heat semigroup on a manifold the limit of the heat semigroups associated to a compact exhaustion?

Let $M$ be a paracompact Riemannian manifold, and $E to M$ a Hermitian vector bundle endowed with a Hermitian connection $nabla$. Write $M$ as an exhaustion $bigcup _{j ge 0} U_j$ with relatively compact open subsets with smooth boundaries.

If $L^2 (M, E)$ is the space of square-integrable sections in $E$, let $L : L^2 (M, E) to L^2 (M, E)$ be the (densely defined) Friedrichs extension of $nabla^* nabla$ (with $nabla^*$ the formal adjoint of $nabla$). By the Hille-Yoshida theorem, this operator generates a $C_0$-semigroup $(mathrm e ^{-tL}) _{t ge 0}$ of contractions in $L^2 (M, E)$.

Similarly, one may perform the same construction on $U_j$ and obtain the operator $L_j$ and the associated semigroup $(mathrm e ^{-tL_j}) _{t ge 0}$ of contractions in $L^2 (M, E_j)$, for every $j ge 0$.

Is there any way in which $mathrm e ^{-tL_j}$ converges to $mathrm e ^{-tL}$ for every $t ge 0$?

I know that this is true when $E = M times mathbb C$ with the usual Hermitian structure, and $nabla$ is just the differential operator acting on smooth functions. The proof, though, relies on the existence of the heat kernel, on its property of being the minimal positive fundamental solution of the heat equation, and on using the monotone convergence theorem (to define the heat kernel of $mathrm e ^{-tL}$ as the pointwise limit of the heat kernels of $mathrm e ^{-tL_j}$). These tools are clearly not available in general bundles, so what to do then? I have tried to use the theorems in chapter VIII of volume 1 of Reed & Simon, but the hypotheses therein are not satisfied in my setting.

fa.functional analysis – Operator Algebra on an Invariant Subset

In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $mathfrak{M}$ of the algebra $mathfrak{A}-mathfrak{L}$ is invariant with respect to the representation $a{rightarrow}A_a^{mathfrak{A}-mathfrak{L}}$. And, from which, it is concluded that $mathfrak{L}_1$ (defined below) is a left ideal in $mathfrak{A}$ that contains $mathfrak{L}$.

I try to arrive at the same conclusion in the following manner. First the representation $a{rightarrow}A_a^{mathfrak{A}-mathfrak{L}}$ means that $mathfrak{A}-mathfrak{L}$ is a left ideal in the algebra $mathfrak{A}$ and that the left regular representation $a{rightarrow}A_a$ of $mathfrak{A}$ is restricted to the $mathfrak{A}-mathfrak{L}$ left ideal (definition according to Rickart). Second, all images of operators in the restricted left representation (image of $x$ denoted by $x^{prime}$) are left ideal images. Third, since $mathfrak{M}$ is an invariant subspace of $mathfrak{A}-mathfrak{L}$ then $mathfrak{L}_1={x,{:},x^{prime},{in},mathfrak{M}}$ must be a left ideal in $mathfrak{A}-mathfrak{L}$ and hence, $mathfrak{L}_1$ is also a left ideal in $mathfrak{A}$. This result is not quite right. According to Rickart I should have, in addition, concluded that $mathfrak{L}_1$ contains $mathfrak{L}$ but this does not seem possible since we’re constrained to $mathfrak{A}-mathfrak{L}$. Can anyone point out what I missed?

Additional question. Rickart makes the statement that $mathfrak{M}$ is a linear subspace of the algebra $mathfrak{A}-mathfrak{L}$. Is he saying that $mathfrak{M}$ is $textit{NOT}$ to be taken as a subalgebra of $mathfrak{A}-mathfrak{L}$ ? Or does he really mean to say that $mathfrak{M}$ is a subalgebra of $mathfrak{A}-mathfrak{L}$ ?

(Sidebar: definition of an invariant subspace. Take $mathfrak{M}$ as a subalgebra of $mathfrak{A}-mathfrak{L}$. If $Tx:{in}:M$ for every $T:{in}:mathfrak{M}$ where $M$ is the set of vectors that generate the algebra $mathfrak{M}$, then $M$ is said to be invariant with respect to the algebra $mathfrak{M}$.)

Rickart: Theorem 2.2.1 page 50 of “$textit{General Theory of Banach Algebras,}$” 1960.

fa.functional analysis – Relationship between various notions of laws of random functions

Let $X,Y,Z$ be locally-compact metric spaces, $f:Xtimes Yrightarrow Z$ be uniformly continuous, and let $xi$ be an $X$-valued random element. Equip the set $C(Y,Z)$ with cylindrical $sigma$-algebra generated by the sets:
$$
C_{B,(y_1,dots,y_n)}:={f:Yrightarrow Z,|f(y_1,dots,y_n) in B},
$$

where $nin mathbb{Z}^+, , y_1,dots,y_nin Y,, Bin mathcal{B}(Z)^{otimes n}$.

What is the relationship between:

  • The law of the $C(Y,Z)$-valued random element $f(xi,cdot)$,
  • The pushforwards ${f(cdot,y)_{#}Law(xi)}_{y in Y}$
  • The pushforward $(xmapsto f(x,cdot))_{#}Law(xi)$?

Where $xmapsto f(x,cdot)$ denotes the map sending an $x in X$ to the function $ymapsto f(x,y)$ in $C(Y,Z)$

fa.functional analysis – Reference Request: Functions with smooth projections on finite-dimensional subspaces

Let $E,F$ be Banach spaces and $F$ be finite-dimensional. Let $fin C(F,E)$ have the property that:
$$
text{For every finite-dimensional subspace $E’subseteq E$ we have } pi_{E’}circ fin C^k(F,E’),
$$

where $pi_{E’}:Erightarrow E’$ is the metric projection. Are such functions studied or characterized anywhere in the literature?

Note: In the case where $E=mathbb{R}^n$ this definition seems to be equivalent to $f in C^k(F,E)$ but I was wondering is this generalization was studied (and where).

fa.functional analysis – The quadratic variation of $int_0^tint_T^Sg(s,x)dW_s^x dx$

Consider the process $W^x_t$ which is a Brownian motion for every $xgeq 0$ such that
$$dlangle W_t^x,W_t^yrangle=Q(x,y)dt$$
where $Q$ is some non-negative definite function. Now consider the integral

$$I(g)=int_0^tint_T^Sg(s,x)dW_s^x dx.$$
How can we find the quadratic variation $langle I(g),I(g) rangle$ of this integral?

My guess is that
$$langle I(g),I(g) rangle=int_0^tint_T^Sint_T^S g(s,x)g(s,y)Q(x,y)dxdydt.$$

However, I do not know if this is correct or how this can be proved (or if this follows from some well-known result). Any help or references are much appreciated!

fa.functional analysis – Nonlinear variation of constants formula

Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), ; x(0)=x_0in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X rightarrow X $ are two nonlinear functions. Furthermore, suppose we know how to solve the unperturbed system $y'(t)=f(y(t)),; y(0)=y_0$, let’s say that $y(t)=phi(y_0,t)$, for some function $phi$ of $t$ and $y_0$. In the case where $dim X<infty,$ one can express the solution $x(t)$ in term of $phi$ via the so called nonlinear variation of constants formula (due to V. M. Alekseev, (1961). “An estimate for the perturbations of the solutions of ordinary differential equations” Westnik Moskov Unn. Ser, 1, 28-36).

Is there an analogous formula to variation of parameters in the case $dim X=+infty$ ?

fa.functional analysis – Density and Fourier Approximation

Let $mathbb{T}$ denote the 1-d torus and $H^s(mathbb{T})$ the Sobolev space of order $sgeq0$ of complex-valued functions on $mathbb{T}$ with the identification $H^0 (mathbb{T}) = L^2 (mathbb{T})$. Let $mathcal{T}_N$ denote the span of the first $N in mathbb{N}$ trigonometric polynomials and $P_N : L^2(mathbb{T}) to mathcal{T}_N$ the standard Fourier projection. It is easily shown, by passing to sequences, that, for any $f in H^s (mathbb{T})$ with $s geq 1$,

$|f-P_Nf|_{L^2} leq bar{N}^{-s}|f|_{H^s}$

where $bar{N} = sqrt{1 + N^2}$. Consider now the following simple argument. Let $f in H^1(mathbb{T})$ and fix $0 < epsilon < 1$. Since $C^infty (mathbb{T})$ is dense in any $H^s (mathbb{T})$ space, there exists a function $g_epsilon in C^infty (mathbb{T})$ such that

$|f – g_epsilon| < frac{epsilon}{2}.$

Triangle inequality implies

$|g_epsilon |_{H^1} < frac{epsilon}{2} + |f|_{H^1}$.

Since $g_epsilon in C^infty (mathbb{T})$, we have $g_epsilon in H^m (mathbb{T})$ for any $m in mathbb{R}$, and, just to keep this as simple as possible, let’s pick $m=2$. Then

$|g_epsilon – P_N g_epsilon|_{L^2} leq bar{N}^{-2} |g_epsilon |_{H^1} leq bar{N}^{-2} ( frac{epsilon}{2} + |f|_{H^1} ) leq frac{epsilon}{2}$

simply by picking the smallest $N in mathbb{N}$ such that $bar{N} geq left ( frac{4 |f|_{H^1}}{epsilon} right )^{1/2}$.

By triangle inequality, we obtain $|f – P_N g_epsilon|_{L^2} leq epsilon$.

So, we have shown that, given any $f in H^1 (mathbb{T})$ then, for any $N in mathbb{N}$, there exists $g_N in mathcal{T}_N$ such that

$|f – g_N|_{L^2} leq 4 bar{N}^{-2} |f|_{H^1}.$

In fact, we could have done much better and gotten a super-algebraic rate of convergence and, if instead working on $mathbb{T}^d$, avoided the curse of dimensionality. My immediate question about this (apart from is it correct) is how does one reconcile it with that fact that $P_N$ is the optimal $L^2$-projector onto $mathcal{T}_N$. In particular, $P_N f$ should be the best one can do when approximating from $mathcal{T}_N$. This suggests one can do a lot better, provided more regularity is available and is perhaps related to the fact that $H^1$ is relatively compact in $L^2$? Another point of view is that method approximation, particularly $f mapsto g_epsilon$ is discontinuous (there are lower bounds that would imply this but they are for non-linear methods and what bothers me most is that $mathcal{T}_N$ is a linear space). Is the operator of best approximation from $mathcal{T}_N$ in $L^2$ linear or continuous? Any thoughts on this would be helpful. Thanks.

fa.functional analysis – Convergence of the solutions of a system of differential equations

Consider this system of differential equations for $tin(0,infty)$:

$$ frac{d}{dt}x(t) = a(t) + F(x(t), y(t)),$$
$$ frac{d}{dt}y(t) = a(t) + G(x(t), y(t)),$$

with positive initial conditions: $y(0)>0, x(0) >0$, where $a(t)$ is a piecewise continuous function and $F$ and $G$ are some continuous functions.

Fix $T>0$ and consider the interval $I=(0,T)$. Assume that on this interval there exists a sequence of (nice) continuous functions ${a_n(t)}_{n=1}^{infty}$ converging to $a(t)$ in $L^1$ $left(int_{0}^{T} |a(t)-a_n(t)| dt rightarrow 0right)$, such that this system of differential equations:

$$ frac{d}{dt}x_n(t) = a_n(t) + F(x_n(t), y_n(t)),$$
$$ frac{d}{dt}y_n(t) = a_n(t) + G(x_n(t), y_n(t)),$$

with the same initial conditions $x_n(0)=x(0)>0$ and $y_n(0)=y(0)>0$, has positive continuous solutions $x_n(t)>0$ and $y_n(t)>0$ for $t in I$.

$bf{1 -}$ Is this information enough to prove that the solutions $x(t)$ and $y(t)$ of the original ODE system is nonnegative:
$$ x(t) ge 0, quad y(t) ge 0.$$
for $t in I.$?

$bf{2- }$ can we prove from continuity condition of $F$ and $G$, that $x_n(t)$ and $y_n(t)$ converge to $x(t)$ and $y(t)$ pointwise or uniformly or in $L^1$? If not, what conditions on $F$ and $G$ (or any other parameters in this problem) make the statement true?

I also appreciate any paper or book on this subject.