Operator Theory Functional Analysis and Applications Extended Heinz and Jensen type inequalities and rearrangements On some applications of representable functionals of a Banach quasi * algebra Minimality properties of Sturm Liouville problems with increasing affine boundary conditions Scattering spectrum and resonance states completeness for quantum graph with Rashba Hamiltonian Tau functions associated with linear systems Groups of orthogonal matrices all orbits of which generate lattices Invertibility issues for Toeplitz plus Hankel operators and their close relatives K inner functions and K contractions Tight and cover to join representations of semilattices and inverse semigroups Calkin images of Fourier convolution operators with slowly oscillating symbols Inner outer factorization of wide rational matrix valued functions on the half plane Convergence rates for solutions of inhomogeneous ill posed problems in Banach space with sufficiently smooth data A closer look at Bishop operators Products of unbounded Bloch functions Birkhoff James orthogonality and applications: a survey The generalized ∂ complex on the Segal Bargmann space The inverse characteristic polynomial problem for trees A note on the Fredholm theory of singular integral operators with Cauchy and Mellin kernels II A note on group representations determinantal hypersurfaces and their quantizations Algebras generated by Toeplitz Operators on the unit sphere II: Non commutative case d modified Riesz potentials on central Campanato spaces On some consequences of the solvability of the Caffarelli Silvestre extension problem Time dependent approach to the uniqueness of the Sommerfeld solution of the diffraction problem by a half plane On the operator Jensen Mercer inequality A numerical approach for approximating variable order fractional integral operator Langlands reciprocity for C* algebras Compact sequences in quasifractal algebras Dilation theory: a guided tour Riesz Fischer maps Semi frames and Frames in rigged Hilbert spaces Periodic coherent states decomposition and quantum dynamics on the flat torus. This book presents 30 articles on the topic areas discussed at the 30th “International Workshop on Operator Theory and its Applications” held in Lisbon in July 2019. The contributions include both expository essays and original research papers reflecting recent advances in the traditional IWOTA areas and emerging adjacent fields as well as the applications of Operator Theory and Functional Analysis. The topics range from C*–algebras and Banach *–algebras Sturm Liouville theory integrable systems dilation theory frame theory Toeplitz Hankel and singular integral operators to questions from lattice group and matrix theories complex analysis harmonic analysis and function spaces. Given its scope the book is chiefly intended for researchers and graduate students in the areas of Operator Theory Functional Analysis their applications and adjacent fields.