|Date||Nov 23, 2022|
In this talk, we consider the following question: What is a circle companion of the Hardy space of the unit disk with values in the space of all bounded linear operators between separable Hilbert spaces? The question on the circle companion is to usually ask whether for each function h on the unit disk, there exists a ``boundary function" bh on the unit circle such that the mapping h -> bh is an isometric isomorphism between Hardy spaces of the unit disk and the unit circle with values in some Banach spaces. The question on the cases of bounded linear operator-valued functions was unsolved until now. We now construct a new space and then this new space is a circle companion of the Hardy space of the unit disk via a ``strong boundary function".