College of Liberal Arts & Sciences

# Operator Theory Seminar

**Abstract:**

For an arbitrary commuting $d$–tuple $T$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\Delta_{t}(T)$ and we prove that the spectral radius of $T$ can be calculated from the norms of the iterates of $\Delta_{t}(T)$ .

We ﬁrst determine the spectral picture of $\Delta_{t}(T)$ in terms of the spectral picture of $T$; in particular, we prove that, for any $0 ≤ t ≤ 1$, $\Delta_{t}(T)$ and $T$ have the same Taylor spectrum, the same Taylor essential spectrum, the same Fredholm index, and the same Harte spectrum. We then study the joint spectral radius $r(T)$, and prove that ${r(T ) = \operatorname{lim}_{n}\left\|\Delta_{t}^{(n)}(T)\right\|}_{2}(0 < t < 1)$, where $\Delta_t^{(n)}(T)$ denotes the $n$-th iterate of $\Delta_{t}(T)$. For $d = t = 1$, we give an example where the above formula fails.

The talk is based on recent research with Chaﬁq Benhida, Sang Hoon Lee and Jasang Yoon.