**Abstract**

The research project concentrates on the development of the symbolic calculus based on spectral decompositions. The spectral decompositions are understood in a sense of the eigenvectors of a given operator, or in a broader sense of working with bases or frames of Hilbert space

**Overzicht**

In dit project werken we aan de ontwikkeling van de symbolische calculus gebasseerd op spectrale decomposities. Deze spectrale decomposties verstaan we in de zin van eigenvectoren voor een gegeven operator, of in een bredere zin als basissen of frames van Hilbertruimten.

### Background Research

We gave a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifolds by Seeley in 1969, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces.

– Dasgupta A., Ruzhansky M., *Eigenfunction expansions of ultradifferentiable functions and ultradistributions, Trans. Amer. Math. Soc., *368

*(2016),*8481-8498. arxiv, link

Also we analysed the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mappings between these spaces on compact manifolds.

– Dasgupta A., Ruzhansky M.,

*Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations*,

**, 5 (**

*Trans. Amer. Math. Soc. Ser. B.**2018*), 81-101.offprint (open access), link, arxiv

We analysed the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.

– Dasgupta A., Ruzhansky M.,

*Eigenfunction expansions of ultradifferentiable functions and ultradistributions. III. Hilbert spaces and Universality.*arxiv