User andrás bátkai - MathOverflow

Answer by András Bátkai for Algorithm to find exponential map of differential operators acting on function

2013-05-18T18:19:57Z

A partial answer: What you call the "exponential function" is the so-called flow semigroup, see Engel-Nagel, Section II.3.28.

Another reference on the Lie derivative is the monograph by Chicone and Swanson.

Answer by András Bátkai for Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$

2013-05-16T19:25:44Z

To give you a reference: Diestel-Uhl, Vector measures, page 98, Chapter 4, Theorem 1:

$$L^p(\mu,X)^\ast = L^q(\mu,X^\ast)$$

if and only if $X^\ast$ has the Radon-Nikodym property with respect to $\mu$.

Here $\mu$ is a finite measure and $p$ and $q$ as usual. Proof can be read there.

ADDED: It should be noted that $L^q(\mu,X^\ast) \subset L^p(\mu,X)^\ast $ always holds, without any condition on the Banach space $X$. The proof of this is quite the same as in the scalar case.

Answer by András Bátkai for radon-nikodým property of $\ell^\infty$

2013-05-07T18:07:16Z

See Proposition 1.2.9 in Arendt-Batty-Hieber-Neubrander. (I look at the first edition now)

Answer by András Bátkai for The Periodic Schrödinger Group

2013-03-22T19:15:25Z

Heuristically it is what Ryan Budney suggests, but of course it does not make sense as a power series, only for so called analytic vectors (which are dense though in this case).

Usually you define this via the spectral theory of selfadjoint operators, see for example

Weidmann: Linear operators in Hilbert space, Section 7.6.

The construction to show that the spectral theorem in this case reduces to the operator being a Fourier multiplier is described in Chapter 10.

Answer by András Bátkai for On exponential formula

2013-02-12T10:14:22Z

If you have an analytic semigroup (generated by a so-called sectorial operator), then the answer is well-known and yes, even convergence rates are possible. This and generalizations were proved by many, see for example the summary and references in

M. Crouzeix, S. Larsson, S. Piskarev, V. Thomée: The stability of rational approximations of analytic semigroups, BIT 33 (1993), 74-84, Theorem 3.

In general it is possible to obtain convergence rates for "nice" initial values. A nice summary of the existing results and vast generalizations are available in this paper by Alexander Gomilko and Yuri Tomilov.

ADDED: After discussion with Delio, let me mention that the result (in Hilbert spaces) is implicitely contained in

H. FUJITA & A. MIZUTANI, "On the finite element method for parabolic equations. I; Approximation of holomorphic semi-groups," J. Math. Soc. Japan, v. 28, 1976, pp. 749-771.

Answer by András Bátkai for Commutator formula in infinite dimensions

2013-02-09T19:33:31Z

For the unbounded case, see

Kato, Tosio Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups. Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), pp. 185–195, Adv. in Math. Suppl. Stud., 3, Academic Press, New York-London, 1978.

The domain condition is that the square roots of $A$ and $B$ should have a dense intersection, then the convergence holds, to the exponential of the form sum of these operators.

Answer by András Bátkai for When should I publish my results?

2012-10-10T09:19:00Z

Understanding old results and putting them into new perspective has its own very important value. You should not underestimate it.

However, especially for a novice, it is difficult to judge this. Consult your reviewer and ask for advice. If an experienced and well-established mathematician sees value in your work, then you should try to publish it.

Of course it is an important and difficult question where. Not all journals approve of this type of synthetizing papers. Here again, the help of an experienced established colleague is inevitable.

Answer by András Bátkai for Generator of a generated $C_0$ semigroup.

2013-01-29T16:49:36Z

The semigroups you construct is in general only weak-* continuous. Are you looking for so-called implemented semigroups?

See for example this paper.

Answer by András Bátkai for Robin-Laplacian in unbounded domains

2013-01-11T12:31:01Z

For the case $p=2$, I would have a look at this paper by Wolfgang Arend and Mahamadi Warma, and its follow-up papers: Potential Analysis 19: 341–363, 2003.

Answer by András Bátkai for Schauder estimates for higher order linear elliptic operator on manifold

2013-01-07T16:09:00Z

This is probably more of a comment: Section 3.2 in Lunardi's book contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder estimates.

On manifolds, you should be able to extend these results using a finite number of coordinate charts (by compactness) as in these notes. But, this is not a reference...

Answer by András Bátkai for New grand projects in contemporary math

2013-01-07T11:06:56Z

The theory of nonlinear dispersive equations, hyperbolic conservation laws, etc., see

Terece Tao's book on the subject, Jean Bourgains book or Helge Holdens co-authored monographs.

Answer by András Bátkai for Generator of a $C_0$-semigroup restricted to a subspace

2013-01-01T10:32:09Z

I do not think such semigroups have been extensively studied.

I have seen such semigroups (and, more generally, evolution families corresponding to the non-autonomous problem) in

A. Lunardi, M. Geissert, Asymptotic behavior and hypercontractivity in nonautonomous Ornstein-Uhlenbeck equations. J. Lond. Math. Soc. (2) 79 (2009), no. 1, 85--106.

Answer by András Bátkai for Old books still used

2012-12-28T17:09:30Z

Sz. Nagy-Foias: Harmonic Analysis of Operators in Hilbert Space (1970) is a still widely used and lively book (though there is a new updated edition in 2012).

T. Kato's Perturbation Theory book (1967) is also definitely in this category, though there is a 1980 second edition and a 1995 reprint.

Nelson Dunford, Jacob T. Schwartz: Linear Operators (1958,1963, 1971). I still take this book regularly into my hands.

An other reference on differential equations is

J. L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems, 1972. It is still "the" reference.

Answer by András Bátkai for Triangularizing a function matrix with smooth eigenvlaues

2012-12-12T06:56:44Z

This is not a complete answer, but the paper

Kreiss, H. O., Über Matrizen die beschränkte Halbgruppen erzeugen, Math. Scand. 7(1959), 71-81.

contains a lot of results in this direction. Unfortunately, at the moment I do not have access to it to check...

Answer by András Bátkai for Counterpart of Weierstrass theorem

2012-11-30T07:39:49Z

You look for the notion of a pseudocompact space.

Answer by András Bátkai for Easy question on Sobolev spaces

2012-10-25T08:34:26Z

I would like to expand a bit what Delio said. Your question is a bit confusing as it is, but we may assume that you mean that $p$ and $q$ are the parameters representing the derivatives. Then what you need is $$ \|f\|_p \leq C\|f\|_q.$$

If you write out the definition of the Sobolev norm on $S$, then you see immediately that this holds. This implies that the identity map defined on $S$ extends uniquely and continuously to the whole $H^q$. This gives you the embedding. (Strictly speaking you have to verify that this extension is the identity map, but this follows easily from continuity considerations.)

Answer by András Bátkai for nonlinear delay differential equation

2012-10-11T20:24:24Z

As mentioned already by Denis Serre, there is a rich literature investigating delay equations.

If you make an experiment, and fix $\bar{x}=1$, then you see that you need as an initial value the complete past on $[-1,0]$. To play a bit, tak as an initial function the constant function $y(s)=1$ for $s\in[-1,0]$. Then you can calculate the solution explicitly for $x\in[0,1]$, then using this you can calculate the solutuion on $[1,2]$, etc. We see that it is far from being analytic. Hence, no chance for a series sepresentation of a solution.

If you are interested in classical stuff, then Bellman and Cooke is an excellent book. An other good reference is the one by Hale and Verduyn Lunel.

ADDED: If it is a delay equation (i.e., $\bar{x}>0$), then the initial condition has to be a function (you have to know the whole past). Then the iteration procedure I described works always. This gives you a possible approximation formula, most numerical methods also work this way.

You are right about analyticity: series representation does it. Smoothness is a consequence. The example I presented to you is only once differentiable at $x=1$, twice at $x=2$, etc. Hence, cannot be analytic.

Answer by András Bátkai for Boundary condition for a non linear schrodinger equation

2012-10-11T07:43:33Z

Well, this does not really answer your question, but formally, you cannot define the trace for functions having regularity $s<1/2$.

Practically, it means that if you take fractional power of the Dirichlet-Laplace operator of order less than $1/4$, then boundary conditions disappear from the domain.

Answer by András Bátkai for Commuting Linear Operators In Hilbert Spaces

2012-10-11T07:30:05Z

I am far from being an expert, but there is a list of results for special cases in a book by Radjavi and Rosenthal, especially in Chapter 9.

Answer by András Bátkai for ordered exponential of unbounded operators

2012-10-10T19:32:54Z

This is usually called the magnus expansion method and has a nice literature in numerical analysis. Kato also used this method to show the existence of solutions in the hyperbolic case.

I would say that strong resolvent continuity and a sufficiently big common domain is sufficient in your case. See Section 5.3 in Pazy.

I can also give a related self-reference, where also the investigation of this product appears and some of the ideas are explained in a simpler situation.

ADDED: My answer concentrates on the method you propose to converge to the solution. To make the content of the references short: yes. A common dense domain and continuity of hte maps $t\mapsto A_tx$ implies the convergence of the product to the solution of the differential equation.

Answer by András Bátkai for Reference request: parabolic PDE

2012-10-07T08:02:41Z

If you are interested in sup-norm and Hölder estimates, then Lunardi's book is a good start:

http://www.amazon.com/Semigroups-Regularity-Parabolic-Differential-Applications/dp/3764351721/ref=la_B001K6J69O_1_1?ie=UTF8&qid=1349596684&sr=1-1

Otherwise you should specify what type of equation are you interested in.

ADDED: Afret the comment of @Liviu:

You should not omit Krylov's books from your list: the one on Hölder spaces and the one on $L^p$ spaces.

And of course there is Evans. An excellent introduction.

ADDED: After the clarification in the question: Topping's lecture notes (there is also a book version from the London Mathematical Society) are quite nice and readable.

Answer by András Bátkai for reference for weak*-semigroup

2012-10-07T16:11:13Z

Echoing the remark of @Bill Johnson, one possibility is van Neerven's book on adjoint semigroups.

Answer by András Bátkai for Reference Request: Schauder theory for fourth-order parabolic equations

2012-10-07T15:58:31Z

I do not have access to the book you cite, but Section 3.2 in Lunardi's book contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder estimates.

Answer by András Bátkai for Must Neuman Elliptic operator has discrete spectrum ?

2012-05-03T14:59:20Z

Of course, on a general domain, the question os how do you define the Neuman Laplacian. There is an excellent exposition in

W. Arendt, A.F.M. ter Elst: Sectorial forms and degenerate differential operators

suggesting methods how to do it.

Answer by András Bátkai for The exponent of self-adjoint operator

2012-04-27T09:27:23Z

Yes. This follows immediately from the spectral theorem and form the functional calculus of selfadjoint operators, even for a much wider range of functions.

Answer by András Bátkai for Orthogonality in non-inner product spaces

2012-03-03T19:24:58Z

Well, it depends what do you need it for. You may also have a look at semi-inner-product spaces, which are natural generalizations of inner product spaces.

Answer by András Bátkai for Analytic generator

2012-02-18T22:45:21Z

Arendt-Batty-Hieber-Neubrander: Vector valued Laplace transforms and Cauchy Problems, First Edition, Examle 3.7.6. The Gaussian semigroup. The proof is the same using Fourier multipliers using the excplicite convolution form of the semigroup.

Where you find a difference is the question whether the domain of the Laplace is a classical function space. This is only true if $p>1$, then you get a Sobolev space. For $p=1$ the domain is strictly bigger than $W^{2,1}(\mathbb{R}^n)$ for $n>1$.

Answer by András Bátkai for Reference request: Simple facts about vector-valued Sobolev space

2012-02-05T20:53:21Z

Herbert Ammann's book on parabolic problems contains an excellent introduction.

Answer by András Bátkai for Reference for Neumann-Laplacian

2011-10-18T22:13:58Z

Have a look at Lunardi's book, it is more on the functional analytic questions you have:

http://books.google.co.uk/books/about/Analytic_semigroups_and_optimal_regulari.html?id=mWojiHzg9bEC

Quanitative de Moivre–Laplace theorem (reference request)

2011-09-29T19:46:24Z

The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution:

$${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)^2 / (2npq)}.$$

My question is: are there more precise, quantitative versions of this theorem in the literature? Are there good estimates how to measure the error? I am unfortunately not familiar with the subject but need a result of this type.

Of course there is always the option of going through existing proofs and checking the details, and turning them from "soft" to "hard", but I suspect this has to be already done. And maybe this is not optimal, maybe there are good accessible ways.

Can someone point me a good reference in this direction?

Comment by András Bátkai

2013-05-22T20:56:07Z

@James Propp: meta.mathoverflow.net/discussion/1296/2/…

Comment by András Bátkai

2013-05-20T19:47:00Z

Sorry, I did not get this comment somehow... Yes, it seems to me as easy as you write.

Comment by András Bátkai

2013-05-20T19:45:17Z

Comment by András Bátkai

2013-05-19T17:49:09Z

The formula for the action on $f(x,y)$ is given in detail in the Engel-Nagel reference below. Does it help?

Comment by András Bátkai

2013-05-18T12:33:06Z

Dear Rafa, it seems that some of your formulae is incomplete, something is missing.And, in particular, I miss your question...

Comment by András Bátkai

2013-05-17T22:17:19Z

A related question: mathoverflow.net/questions/130276/…

Comment by András Bátkai

2013-05-17T19:44:12Z

As mentioned by Lee, math.stackexchange.com is a perfect place.

Comment by András Bátkai

2013-05-17T12:46:51Z

I am a bit slow, why is the left hand side infinite?

Comment by András Bátkai

2013-05-16T19:09:03Z

I am a bit confused. If $X$ is a Hilbert space, then $L^2(0,T;X)$ is a Hilbert space (complete + norm comes from a scalar product). Hence if you also identify $X$ with its dual (as you do with $L^2$), then the statement follows. Maybe this is not what you ask?

Comment by András Bátkai

2013-05-14T12:49:12Z

Have you tried to ask it here: scicomp.stackexchange.com ?

Comment by András Bátkai

2013-05-11T21:25:04Z

Your point 2 is not the same as above: it follows immediately from the continuity of the norm.

Comment by András Bátkai

2013-05-11T21:23:00Z

Though I believe this question is better suited at math.stackexchange.com , let me give you a hint. How do you define your metrics on compact sets?

Comment by András Bátkai

2013-05-10T17:57:21Z

Volume 1: rd.springer.com/book/10.1007/978-3-642-65161-8/… , but there are three.

Comment by András Bátkai

2013-05-10T17:55:02Z

A standard reference on this is the monograph by Lions and Magenes: Non-Homogeneous Boundary Value Problems and Applications. Everyone refers to it for the proof...

Comment by András Bátkai

2013-05-10T10:18:47Z

Corollary 2.2 here: math.psu.edu/bressan/PSPDF/sobolev-notes.pdf