A First Course in Functional Analysis – a second, corrected printing is now available

And now, a word from our sponsor (me):

My textbook for undergraduate functional analysis – A First Course in Functional Analysis – is now available in a second, corrected printing. There were several annoying mistakes or typos that I found about during the five years since it appeared (here is a list of the most substantial corrections). If I understand correctly how this works: people who have access to the e-book through their library will get to the corrected version. If your library does not yet have hard copies, then now would be a great time to order them. Hard copies – I mean real books – are the best! Just take a look at it. If you are teaching a course in functional analysis you can ask the publisher for an inspection copy. Here is the preface of the book if you want to get a feeling of what kind of book it.

Seminar talk by Hartz: How can you compute the multiplier norm?

Happy new year!

Next Thursday, January 7th, 2021, Michael Hartz will speak in our Operator Algebras and Operator Theory seminar.

Title: How can you compute the multiplier norm?

Time: 15:30-16:30

Zoom link: Email me.

Abstract:

Multipliers of reproducing kernel Hilbert spaces arise in various contexts in operator theory and complex analysis. A basic example is the Hardy space , whose multiplier algebra is , the algebra of bounded holomorphic functions. In particular, the norm of a multiplier on is the pointwise supremum norm. 

For general reproducing kernel Hilbert spaces, the multiplier norm can be computed by testing positivity of matrices analogous to the classical Pick matrix. For , suffices. I will talk about when it suffices to consider matrices of bounded size . Moreover, I will explain how this problem is related to subhomogeneity of operator algebras.

This is joint work with Alexandru Aleman, John McCarthy and Stefan Richter

Seminar talk at the BGU OA Seminar

This coming Thursday (July 2nd, 14:10 Israel Time) I will be giving a talk at the Ben-Gurion University Math Department’s Operator Algebras Seminar. If you are interested in a link to the Zoom please send me an email.

I will be talking mostly about these two papers of mine with co-authors: older one, newer one. Here is the title and abstract:

Title: Matrix ranges, fields, dilations and representations

Abstract: In my talk I will present several results whose unifying theme is a matrix-valued analogue of the numerical range, called the matrix range of an operator tuple. After explaining what is the matrix range and what it is good for, I will report on recent work in which we prove that there is a certain “universal” matrix range, to which the matrix ranges of a sequence of large random matrices tends to, almost surely. The key novel technical aspects of this work are the (levelwise) continuity of the matrix range of a continuous field of operators, and a certain quantitative matrix valued Hahn-Banach type separation theorem. In the last part of the talk I will explain how the (uniform) distance between matrix ranges can be interpreted equivalently as a “dilation distance”, which can be interpreted as a kind of “representation distance”. These vague ideas will be illustrated with an application: the construction of a norm continuous family of representations of the noncommutative tori (recovering a result of Haagerup-Rordam in the d=2 case and of Li Gao in the d>2 case).

Based on joint works with Malte Gerhold, Satish Pandey and Baruch Solel.

A review of my book A First Course in Functional Analysis

A review for my book A First Course in Functional Analysis appeared in Zentralblatt Math – here is a link to the review. I am quite thankful that someone has read my book and bothered to write a review, and that zBMath publishes reviews. That’s all great. Now I have a few words to say about it. This is an opportunity for me to bring up the subject of my book and highlight some things worth highlighting.

I am not too happy about this review. It is not that it is a negative review – actually it has a rather kind air to it. However, I am somewhat disappointed in the information that the review contains, and I am not sure that it does the reader some service which the potential readers could not achieve by simply reading the table of contents and the preface to the book (it is easy to look inside the book in the Amazon page; of course, it is also easy to find a copy of the book online).

The reviewer correctly notices that one key feature of the book is the treatment of as a completion of , and that this is used for applications in analysis. However, I would love it if a reviewer would point out to the fact that, although the idea of thinking about as a completion space is not new, few (if any) have attempted to actually walk the extra mile and work with in this way (i.e., without requiring measure theory) all the way up to rigorous and significant applications in analysis. Moreover, it would be nice if my attempt was compared to other such attempts (if they exist), and I would like to hear opinions about whether my take is successful.

I am grateful that the reviewer reports on the extensive exercises (this is indeed, in my opinion, one of the pluses of new books in general and my book in particular), but there are a couple of other innovations that are certainly worth remarking on, and I hope that the next reviewer does not miss them. For example, is it a good idea to include a chapter on Hilbert function spaces in an introductory text to FA? (a colleague of mine told me that he would keep that out). Another example: I think that my chapter on applications of compact operators is quite special. This chapter has two halves: one on integral equations and one on functional equations. Now, the subject of integral equations is well trodden and takes a central place in some introductions to FA, and one might wonder whether anything new can be done here in terms of the organization and presentation of the material. So, I think it is worth remarking about whether or not my exposition has anything to add. The half on applications of compact operators to functional equations contains some beautiful and highly non-trivial material that has never appeared in a book before, not to mention that functional equations of any kind are rarely considered in introductions to FA; this may also be worth a comment.