Talks

Suppose that $A$ is a bounded linear operator on the Hardy space $H^p$ that satisfies
\[\langle Az^j,z^k \rangle = a_{k-j} \quad (j,k \in \mathbb{N}_0)\]
for some sequence of complex numbers $\{a_n\}_{n \in \mathbb{Z}}$. By the Brown--Halmos theorem, $A$ must be a Toeplitz operator with bounded symbol, that is, $\{a_n\}_{n \in \mathbb{Z}}$ is the Fourier sequence of a bounded function. Likewise, Nehari's theorem shows that if $A$ satisfies $\langle Az^j,z^k \rangle = a_{k+j+1}$ instead, then $A$ is equal to a Hankel operator with bounded symbol. These results were proven in the 50's and 60's and have become classical in the theory of Hardy spaces.

More recently, due to some applications in mathematical physics, there has been a lot of interest in so-called Toeplitz+Hankel operators. Quite simply put, a Toeplitz+Hankel operator is the sum of a Toeplitz operator $T(a)$ and a Hankel operator $H(b)$. Now clearly, if both $T(a)$ and $H(b)$ are bounded, then $A = T(a)+H(b)$ is necessarily bounded as well. It is therefore natural to ask whether the converse is also true or if the ``unboundedness'' of $T(a)$ and $H(b)$ can somehow cancel out. I will elaborate on this question and present a Brown--Halmos type result for Toeplitz+Hankel operators for both the Hardy spaces $H^p$ and the sequence spaces $\ell^p(\mathbb{N}_0)$. A similar characterization for compactness will be obtained as well.

Based on joint work with Torsten Ehrhardt and Jani Virtanen.

Hallo liebe Institutsmitarbeiter*innen,

anbei der offizielle Einladungstext zum Promotionsvortrag von Alexander Haupt:

_________________________________________________________________________

Sehr geehrte Damen und Herren,

im Rahmen seines Promotionsverfahrens wird

Herr M. Sc. Alexander Michael Haupt

einen kombinierten Live-Online-Vortrag mit dem Titel

„New Combinatorial Proofs for Enumeration Problems and Random Anchored Structures“

halten. Der Vortrag findet statt am

Dienstag, dem 21. September 2021 um 11:00 Uhr.

Zu diesem universitätsöffentlichen Vortrag lade ich Sie herzlich ein.

Aufgrund der aktuell geltenden Regelungen können Interessierte nur per Zoom am Vortrag teilnehmen. Bitte benutzen Sie hierzu die folgenden Zugangsdaten:

https://tuhh.zoom.us/j/88353220627?pwd=amZFYjRNOHl2TkdGb2c1Z29MVGNCUT09

Meeting-ID: 883 5322 0627
Kenncode: 747604

Mit freundlichen Grüßen
Prof. Dr. Matthias Schulte

(Vorsitzender des Prüfungsausschusses)