Under a suitable ellipticity condition, we show that classical SG-pseudodifferential operators of nonnegative order possess complex powers. We show that the powers are again classical and derive an explicit formula for all homogeneous components.

We obtain a Weyl formula for a class of positive md-elliptic operators on manifolds with finitely many cylindrical exits. Weyl formula is deduced by a classical Tauberian theorem through the asymptotic expansion at t = 0 of the trace of the heat parametrix. The constant of the leading term is expressed invariantly by means of the usual principal symbol and exit symbols.

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG-pseudodifferential operators on R n and suitable manifolds, using a finite-part integral regularization technique. This allows us to define a zeta-regularized determinant det A for parameter-elliptic operators A ∈ S

Let M be a closed manifold. We show that the Kontsevich-Vishik trace, which is defined on the set of all classical pseudodifferential operators on M , whose (complex) order is not an integer greater than or equal to − dim M , is the unique functional which (i) is linear on its domain, (ii) has the trace property and (iii) coincides with the L 2 -operator trace on trace class operators.Also the extension to even-even pseudodifferential operators of arbitrary integer order on odd-dimensional manifolds and to even-odd pseudodifferential operators of arbitrary integer order on even-dimensional manifolds is unique. MSC 2000: 58J40, 58J42, 35S05

We deal with the asymptotic behaviour for λ → +∞ of the counting function N P (λ) of certain positive selfadjoint operators P with double order (m, µ), m, µ > 0, m µ, defined on a manifold with ends M. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier Integral Operators associated with weighted symbols globally defined on R n . By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for N P (λ) and show how their behaviour depends on the ratio m µ and the dimension of M.2010 Mathematics Subject Classification. Primary: 58J40; Secondary: 35S05, 35S30, 47G30, 58J45.