Abstracts

We discuss a kind of spectral theory for bounded linear operators $T$ on a Hilbert space $H$ satisfying\begin{equation}\label{eq}
\alpha(T^*,T) := \sum_{n=0}^{\infty} \alpha_n T^{*n} T^n \ge 0 \quad \text{(convergence in SOT)},\end{equation} where $\alpha(t) = \sum \alpha_n t^n$ is an analytic function with $\alpha_n \in \mathbb{R}$ and $\alpha_0 = 1$. This type of conditions has been extensively analysed, starting from pioneering work by Agler in the 80's. There are also many papers on tuples of commuting operators.

Put $k(t)=1/\alpha(t) = \sum k_n t^n$. In 2018, Bickel, Clouâtre, Hartz and McCarthy proved (even for tuples of commuting operators) that if $\alpha_n \lt 0$ for every $n \ge 1$ and the quotients $k_{n+1}/k_n$ tend to $1$ then \eqref{eq} holds if and only if $T$ extends to $(B_k \otimes I_\mathcal{R})\oplus S$, where $B_k$ is a backward shift on the RKHS of analytic functions associated to $k$, $\mathcal{R}$ is an auxiliary Hilbert space and $S$ is an isometry. As a complement of this result, we show that $(V_D,W)$ provides a minimal model of $T$, where $V_D$ is a contraction given by \[V_D x (z) := D (I-zT)^{-1} x, \quad D := (\alpha(T^*,T))^{1/2}\] and $W:= (I_H - V_D^* V_D)^{1/2}$. So $D$ plays the role of the defect operator of $T$.

Let $\mathcal{A}_T$ be the set of analytic functions $\alpha$ with summable Taylor coefficients such that $\sum |\alpha_n| T^{*n}T^n$ converges in SOT and let $\mathcal{A}_T^0$ be the closure of the polynomials in $\mathcal{A}_T$. Define \[ \mathcal{C}_\alpha^w := \{ T \in L(H) \, : \, \alpha \in \mathcal{A}_T, \alpha(T^*,T) \ge 0 \}.\]

Using Gelfand Theory and a factorization lemma for polynomials, we prove the following result. Let $T \in L(H)$ with $\sigma(T) \subset \overline{\mathbb{D}}$. Suppose that $(1-t)^a \in \mathcal{A}_T$, where $a\gt 0$, and let $\beta\in \mathcal{A}_T^0$ satisfy that $\beta(t)\gt 0$ for every $t \in [0,1]$. If \[ \alpha(t) = (1-t)^a \beta(t)\] and $T \in \mathcal{C}_\alpha^w$, then $T$ is similar to an operator in $\mathcal{C}_{(1-t)^a}^w$.

This result applies to many functions $\alpha$ such that $k(t)$ has negative coefficients and Agler’s techniques do not work. We will give some consequences for Ergodic Theory. In particular, we show that any operator in $\mathcal{C}^w_{(1-t)^a}$ is quadratically $(C,b)$-bounded whenever $0 \lt 1-a \lt b$.

If time permits, we will also discuss a new model of an operator in an annulus. In this case, the annulus turns out to be a complete $K$-spectral set of the operator.

This is a joint work in progress with Luciano Abadias and Dmitry Yakubovich.

Recall that, if a subset $\Omega$ of the complex plane contains the numerical range of a bounded operator $A$ on a Hilbert space $H$, then $\Omega$ is a $C(\Omega)$-spectral set for $A$, i.e. $ \|f(A)\|\leq C(\Omega)\sup_{z\in \Omega}|f(z)|, $ for all rational functions $f$ bounded in $\Omega$. I have made the conjecture that $C(\Omega)\leq 2$ and nowadays the best estimate (due to César Palencia) is $C(\Omega)\leq 1{+}\sqrt2$. I will speak about this estimate and propose some variations allowing to consider non convex situations.

This talk is based on a collaboration with Anne Greenbaum.

A well known result due to Carleson characterizes interpolating sequences for scalar valued functions in $\mathrm{H}^\infty(\mathbb{D})$. In this talk we extends his theorem to sequences of matrices, with no assumptions on their sizes. In particular, we will relate an interpolating problem for a sequence of matrices to a quasi free interpolating problem for their spectra.

Let $A$ be a complex Banach algebra with identity $1$, $a\in A$, $n\in \mathbb{Z}_+$ and $\epsilon>0$. The $(n,\epsilon)$-pseudospectrum $\Lambda_{n,\epsilon}(A, a)$ of $a$ is defined as \begin{align*}{\Lambda_{n,\epsilon} (A,a):=\left\{\lambda \in \mathbb{C}: (\lambda-a) \text{ is not invertible or } \|(\lambda-a)^{-2^{n}}\|^{1/2^n} \geq \frac{1}{\epsilon}\right\}}.\end{align*}

Some elementary properties of $\Lambda_{n,\epsilon}(A,a)$ will be discussed. If $p\in A$ is a non-trivial idem-potent, then the subalgebra $pA p$ is a Banach algebra, called reduced Banach algebra with identity $p$. Suppose $q=1-p$. For $a\in A$ with $ap = pa$, we examine the relationship between the $(n,\epsilon)$-pseudospectrum of $a$, $\Lambda_{n,\epsilon}(A, a)$ and $(n,\epsilon)$ -pseudospectra of $pap \in pA p$, $\Lambda_{n,\epsilon}(pA p, pap)$ and of $qaq \in qA q$ , $\Lambda_{n,\epsilon}(q A q, qaq)$. We note that in the set-up of operators on Banach spaces, this is equivalent to an operator $T$ having a block diagonal representation with respect to a particular direct sum decomposition. We also extend this study by considering a finite number of idempotents $p_1 ,\cdots, p_n$, as well as an arbitrary family of idempotents satisfying certain conditions.

This is based on a joint work with S. H. Kulkarni, Professor, Indian Institute of Technology Madras, Chennai.

Functional version for Furuta parametric relative operator entropy is studied. Some inequalities are also discussed. In addition, we introduce the Heron and Heinz means of two convex functionals. Some inequalities involving these functional means are also investigated. The operator versions of our theoretical functional results are immediately deduced. We also obtain new refinements of some known operator inequalities via our functional approach. The theoretical results obtained by our functional approach immediately imply those of operator versions in a simple, fast and nice way. This is joint work with M. Raissouli.

I generalized several Berezin number inequalities involving  product of operators, which acting on a Hilbert space $\mathscr H(\Omega)$. Among other inequalities, it is shown that if $A, B$ are positive operators and $X$ is any operator, then
\begin{align*}
\textbf{ber}^{r}(H_{\alpha}(A,B)&)\leq\frac{\|X\|^{r}}{2}\left(\textbf{ber}(A^{r}+B^{r})-2\inf_{\|\hat{k}_{\lambda}\|=1}\eta(\hat{k}_{\lambda})\right)\\&
\leq\frac{\|X\|^{r}}{2}\Big(\textbf{ber}(\alpha A^{r}+(1-\alpha)B^{r})+\textbf{ber}((1-\alpha)A^{r}+\alpha B^{r})\\&\qquad\qquad\qquad-2\inf_{\|\hat{k}_{\lambda}\|=1}\eta(\hat{k}_{\lambda})\Big),
\end{align*}
where $\eta(\hat{k}_{\lambda})=r_{0}(\langle A^{r}\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle^{\frac{1}{2}}-\langle B^{r}\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle^{\frac{1}{2}})^{2}$, $r\geq 2$, $0\leq\alpha\leq1$, $r_{0}=\min\{\alpha,1-\alpha\}$ and $H_{\alpha}(A,B)=\frac{A^\alpha XB^{1-\alpha}+A^{1-\alpha} XB^{\alpha}}{2}$.

A classical result of Sz.-Nagy and Foias shows that every contraction $T$ on a Hilbert space without unitary summand admits an $H^{\infty}$-functional calculus, that is, one can make sense of $f(T)$ for every bounded analytic function f in the unit disc. In multivariable operator theory, one studies tuples of commuting operators instead of the single operator $T$ . In this setting, the role of $H^{\infty}$ is played by multiplier algebras of certain reproducing kernel Hilbert spaces. I will talk about a generalization of the Sz.-Nagy--Foias functional calculus, which applies to a large class of multiplier algebras on the unit ball in $\mathbb{C}^d$. This is joint work with Kelly Bickel and John McCarthy.

Markov-type inequalities provide upper bounds on the norm of the (higher order) derivative of an algebraic polynomial in terms of the norm of the polynomial itself. The investigation for classical $L^2$-norms began in the 1940s with Erhardt Schmidt. Only recently, results for higher order derivatives in cases where the norms are of the Laguerre, Gegenbauer, or Hermite type, were found by Böttcher and Dörfler.

The results were extended to inequalities where the norms on both sides of the inequality may be chosen differently. We will show how this problem can be treated for other norms and hint on some problems that arise on the way.

The Oka extension theorem says that if $p=(p_1, \dots, p_m)$ is an $m$-tuple of polynomials in $d$ variables, and $f$ is a function holomorphic on a neighborhood of the polynomial polyhedron $\{ z \in {\mathbb C}^d : |p_j (z) | \leq 1, 1 \leq j \leq m \}$, then there is a function $F$ holomorphic on a neighborhood of the polydisk in $d+m$ variables so that $f(z) = F(z, p(z))$.

If one asks for norm bounds on $F$ in terms of  $f$, one is led to consider function norms defined in terms of evaluating the functions on certain $d$-tuples of commuting operators. We will discuss this, and show how this approach leads to a reformulation of the Crouzeix conjecture to a conjecture about extending holomorphic  functions from certain subvarieties of the polydisk to the whole polydisk.

This is joint work with Jim Agler and Nicholas Young.

It is known that, in a Hilbert space setting,  the numerical range is a $(1+\sqrt{2})$-spectral set. This means that for any linear operator $A : H \to H$, acting on a Hilbert space $H$, and for any holomorphic mapping $f$, defined at least on the numerical range $W(A)$ of a $A$,  there holds $$ \| f(A) \| \le (1+\sqrt{2}) \sup_{z \in W(A)} | f(z)|.$$

After reconsidering different alternatives of the proof of this result, an improvement of it is presented.

Crouzeix observed in 2007 that for any operator $A$ in a Hilbert space and any polynomial $p$ $\|p(a)\|\leq C\sup_W|p(z)|$, where $W$ is the numerical range of $A$ and the constant $C$ is universal, i.e. does not depend neither on the operator nor on the space. He also proved in the same paper that $2\leq C\leq 11.08$ and conjectured that $C=2$. We will review recent developments on proving the conjecture ($C\leq 1+\sqrt{2}$) and show some deformations of the numerical range that may lead to new constants. (joint work with P. Pagacz and M. Wojtylak)

We consider Fock representations of the generalized commutation relations of the form $a(s)a^+(t)=Q(s,t)a^+(t)a(s)+d(s,t)$, where $s,t $ are in $R^n, |Q(s,t)|\leq1$ and  $d(s,t)$ is defined by smeared double integral. This contains the anyon statistics with $|Q(s,t)|=1$. The operators $a(t)$ and $a^+(t)$ are realized as (distribution-valued) creation and annihilation “at point” on a $Q$-deformed Fock space. Additional relations of the form $a(s)a(t)=Q(t,s)a(t)a(s)$ are obtained for these $s,t$ for which $ |Q(s,t)|=1$.

The talk is based on a joint paper: Bozejko, Lytvynov, Wysoczanski, Fock representations of Q-deformed commutation relations, Journal of Mathematical Physics 58, (2017).