My research interest lies in some interaction areas of functional analysis and applied harmonic analysis,  where operator algebra/operator theory, representation theory and frames theory are very much involved. Some of my earlier research was focused on the theory of (non self-adjoint) operator algebras, foundations of frame theory including operator theoretical approach to wavelet and Gabor analysis, and fractal measures. My relatively recent research focuses include: 

I. Dilation theory 

This is a long term project with the goal of classifying operator valued measures (both on commutative domain and non-commutative domain — projections in a von Neumann algebras) from the dilation point of view. This project was initially motivated by the dilations of frames, which can be considered as a special case of an OVM. There is corresponding theory for bounded linear maps. While the Hilbertian dilations usually associated with complete boundedness of the OVM or the linear map, Banach space theory plays an essential role in establishing a general dilation theory for arbitrary OVM/maps. The theory of quantum measures can be viewed as a non-commutative frame theory since their domain spaces contains non-commutative objects such as quantum states (projections) from a non-commutative von Neumann algebra. A general dilation theory for quantum measures need to be established. A central problem asks about the existence of normal dilations for Banach space operator valued normal maps on arbitrary von Neumann algebras. There is a good chance of establishing a class of (possibly new and different from the traditionally studied) non-commutative L^p-spaces built on the  space of OVMs with bounded total p-variations.

II. Frame theory aspects of representation theory

This project aims at exploring some intrinsic connections involving frame theory and representation theory. In other words, I am generally interested in the problems related to (mostly) theoretical aspects of the so-called structured frame theory.

• Duality theory for projective frame representations: A central problem is to characterize the groups such that every frame representation admits a commutant dual pair. This question has a surprising connection with the classification problem of free group algebras
• Phase retrievable frames: Characterize/construct various type of frames that can be used to perform phase-retrieval for various signal spaces, i.e., recover rank one projection (pure state) from the magnitudes of the from coefficients
• Phase retrievability of representation frames: Investigate phase retrieval property of the frames induced by group representations. An open problem is: Does every irreducible projective unitary representation of any finite non-abelian group admit a single vector such that its orbit is a phase-retrievable frame. While lots of work has been done for phase-retrievable Gabor frames, very little is known for general phase-retrievable  representation frames
• Frame theory in mathematical foundations of QIT:  Examine various type of  quantum channels in connection with frame theory, in particular, covariant quantum channels, targeted fidelity preserving,  entanglement breaking, mixed unitary channels, spectral theory for generalized twirling maps

III. Dynamical frames and samplings (part of the structured frame theory)

The theory of dynamical frames evolved from practical problems in dynamical sampling where the initial state of a vector needs to be recovered or determined from the space-time samples of evolutions of the vector. This leads to the investigation of structured frames obtained from the orbits of (evolution) operators. Since many interested cases involve semigroup representations, it naturally leads to the question of establishing the connections of dynamical sampling theory with the theory of semigroup algebras (non-self-adjoint operator algebras).

• Interested in the classification of frame generators in the dynamical settings, e.g. involving the central frame representations of groups,   evolution semigroups, and the hybrid case.
• Interested in Building connections of frame representation theory with Beurling and Sarason’s Theory for shift invariant and hperinvariant subspaces of Hardy spaces over polydiscs, and possibly their non-commutative analogs over non-abelian (e.g. free) semigroups.