Student APS
Speaker: Swapan Jana, IIT Bombay
Host: Sourav Pal
Title: ANDOˆ ’S DILATION AND BOHR-THEORETIC ASPECTS OF THE MULTIVARIATE VON NEUMANN INEQUALITY FOR BANACH SPACE OPERATORS
Time, day and date: 12:00:00 PM – 1:00:00 PM, Wednesday, February 18
Venue: Room 105
Abstract: The celebrated Bela Sz.-Nagy dilation theorem states that every contraction ´ T on a Hilbert space H dilates to an isometry and the minimal isometric dilation space is uniquely determined (up to unitary equivalence) by H ⊕2 `2(DT ), which equals H ⊕2 `2(H ) if T is a strict contraction. In 1963, Ando proved that a commuting pair of contractions on a Hilbert space ˆ H dilates to a commuting pair of isometries on H ⊕2 `2(H ⊕2 H ⊕2 H ⊕2 H ). We show that a commuting pair of strict contractions on a Hilbert space H admits an isometric dilation on H ⊕2 `2(H ⊕2 H ). Moreover, we prove that H ⊕2 `2(H ⊕2 H ) is a minimal isometric dilation space, which shows that minimal isometric dilation space is independent of the choice of the commuting pair of strict contractions on the underlying Hilbert space. An immediate application of our construction is that we extend this mechanism to construct an isometric dilation of a commuting pair of strict contractions on a Banach space. This part is based on a joint work with Prof. Sourav Pal. For c ∈ C n (n ≥ 1), let Gc be the collection of tuples (T1,...,Tn) of Banach space operators satisfying kTik ≤ |ci | for 1 ≤ i ≤ n. Using a multidimensional analogue of Bohr’s theorem, we show that the closed polydisk D n (R) is a spectral set for Gc if and only if c ∈ B(D n (R)), the Bohr zone of D n (R). In contract to the one-variable case, we prove that D n (R) is not a minimal spectral set of Gc for any c ∈ B(D n (R)) when n ≥ 2. This part is based on an ongoing joint work with Prof. Sourav Pal and Dr. Himadri Halder.