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A pseudodifferential calculus for parameter-dependent operators on smooth manifolds with boundary in the spirit of Boutet de Monvel's algebra is constructed. The calculus contains, in particular, the resolvents of realizations of differential operators subject to global projection boundary conditions (spectral boundary conditions are a particular example); resolvent trace asymptotics are easily derived. The calculus is related to but different from the calculi developed by Grubb and Grubb-Seeley. We use ideas from the theory of pseudodifferential operators on manifolds with edges due to Schulze, in particular the concept of operator-valued symbols twisted by a group-action. Parameter-ellipticity in the calculus is characterized by the invertibility of three principal symbols: the homogeneous principal symbol, the principal boundary symbol, and the so-called principal limit symbol. The principal boundary symbol has, in general, a singularity in the co-variable/parameter space, the principal limit symbol is a new ingredient of our calculus., Comment: 75 pages

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn--Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in $L^p(\mathbb R^n_+)\times L^p(\mathbb R^{n-1})$., Comment: 41 pages

Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded $H_{\infty}$-calculus in appropriate $L_{p}$-Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation., Comment: 31 pages

Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb-Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential oerators originally proved by Grubb-Seeley. Another application is the invertibility of parameter-dependent operators of Toeplitz type, i.e., operators acting in subspaces determined by zero-order pseudodifferential idempotents., Comment: Completely worked over and extended; title changed; 42 pages

We introduce a calculus for parameter-dependent singular Green operators on the half-space $\mathbb{R}^n_+$ that combines both elements of Grubb's calculus for boundary value problems of finite regularity and techniques of Schulze's calculus for pseudodifferential operators on manifolds with edges., Comment: 41 pages

We prove that parameter-elliptic extensions of cone differential operators have a bounded $H_\infty$-calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.

We investigate the $\mathcal R$-boundedness of operator families belonging to the Boutet de Monvel calculus. In particular, we show that weakly and strongly parameter-dependent Green operators of nonpositive order are $\mathcal R$-bounded. Such operators appear as resolvents of non-local (pseudodifferential) boundary value problems. As a consequence, we obtain maximal $L_p$-regularity for such boundary value problems. An example is given by the reduced Stokes equation in waveguides.

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 classical parameter-elliptic SG-operators A of order (\mu,m), with \mu>0, m\ge0. For m=0, the asymptotics of TR exp(-tA) as t\to 0 and of TR (\lambda-A)^{-k}$ as |\lambda|\to\infty are derived. For suitable pairs (A,A_0) we show that det A/det A_0 coincides with the so-called relative determinant det(A,A_0)., Comment: Slightly revised; accepted for publication in Math. Nachrichten

We present a calculus of pseudodifferential operators that contains both usual parameter-dependent operators -- where a real parameter \tau\ enters as an additional covariable -- as well as operators not depending on \tau. Parameter-ellipticity is characterized by the invertibility of three associated principal symbols. The homogeneous principal symbol is not smooth on the whole co-sphere bundle but only admits directional limits at the north-poles, encoded by a principal angular symbol. Furthermore there is a limit-family for \tau\to+\infty. Ellipticity permits to construct parametrices that are inverses for large values of the parameter. We then obtain sub-calculi of Toeplitz type with a corresponding symbol structure. In particular, we discuss invertibility of operators of the form P_1A(\tau)P_0 where both P_0 and P_1 are zero-order projections and A(\tau) is a usual parameter-dependent operator of arbitrary order or A(\tau)=\tau^\mu-A with a pseudodifferential operator A of positive integer order \mu., Comment: 22 pages; significant chances in the introduction and Section 4.2; Section 5 worked over

Let L^\star be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T^\star be a subalgebra of operators of the form P_1AP_0, where P_0 and P_1 are two projections. The elements of L^\star act as linear continuous operators in certain scales of abstract Sobolev spaces, the elements of the subalgebra in the corresponding subspaces determined by the projections. We study how the ellipticity in L^\star descends to T^\star, focusing on parametrix construction, Fredholm property, and homogeneous principal symbols. Applications concern SG-pseudodifferential operators, pseudodifferential operators on manifolds with conical singularities, and Boutet de Monvel's algebra for boundary value problems. In particular, we derive invertibilty of the Stokes operator with Dirichlet boundary conditions in a subalgebra of Boutet de Monvel's algebra. We indicate how the concept generalizes to parameter-dependent operators., Comment: 29 pages

We study cone differential operators on the half-axis and edge-degenerate differential operators on a half-space. We construct subspaces of edge Sobolev spaces that can be considered as natural domains for edge-degenerate operators and indicate how they can be used in the study of boundary problems for edge-degenerate operators., Comment: 18 pages

A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a projection under any residual trace depends only on the principal part of the projection. This general, purely algebraic statement applied to the algebra of projective pseudodifferential operators implies that the noncommutative residue factors to a map from the twisted K-theory of the co-sphere bundle. We use arguments from twisted K-theory to show that this map vanishes, thus showing that the noncommutative residue of a projective pseudodifferential projection vanishes. This also gives a very short proof in the classical setting., Comment: 16 pages, LaTeX, third version, new proof of Proposition 3.4, Section 7 removed

We establish the existence of a bounded $H_\infty$-calculus for a large class of hypoelliptic pseudodifferential operators on R^n and closed manifolds., Comment: Final version; accepted for publication in Proc. Amer. Math. Soc

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.

We study closed extensions A of an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted L_p-space. Under suitable conditions we show that the resolvent (\lambda-A)^{-1} exists in a sector of the complex plane and decays like 1/|\lambda| as |\lambda| tends to infinity. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem \dot{u}-\Delta u=f, u(0)=0., Comment: 29 pages