approximation numbers of weighted composition operators we study the approximation numbers of weighted composition operators f mapsto w cdot f circ varphi on the hardy space h 2 on the unit disc for general classes of such operators upper and lower bounds on their approximation numbers are derived for the special class of weighted lens map composition operators with specific weights we show how much the weight w can improve the decay rate of the approximation numbers and give sharp upper and lower bounds these examples are motivated from applications to the analysis of relative commutants of special inclusions of von neumann algebras appearing in quantum field theory borchers triples

an oka principle for equivariant isomorphisms let g be a reductive complex lie group acting holomorphically on normal stein spaces x and y which are locally g biholomorphic over a common categorical quotient q when is there a global g biholomorphism x y if the actions of g on x and y are what we with justification call generic we prove that the obstruction to solving this local to global problem is topological and provide sufficient conditions for it to vanish our main tool is the equivariant version of grauert s oka principle due to heinzner and kutzschebauch we prove that x and y are g biholomorphic if x is k contractible where k is a maximal compact subgroup of g or if x and y are smooth and there is a g diffeomorphism x y over q which is holomorphic when restricted to each fibre of the quotient map x q we prove a similar theorem when is only a g homeomorphism but with an assumption about its action on g finite functions when g is abelian we obtain stronger theorems our results can be interpreted as instances of the oka principle for sections of the sheaf of g biholomorphisms from x to y over q this sheaf can be badly singular even for a low dimensional representation of sl2 our work is in part motivated by the linearisation problem for actions on n it follows from one of our main results that a holomorphic g action on n which is locally g biholomorphic over a common quotient to a generic linear action is linearisable

lattices of jordan algebras commutative jordan algebras play a central part in orthogonal models the generations of these algebras is studied and applied in deriving lattices of such algebras these lattices constitute the natural framework for deriving new orthogonal models through factor aggregation and disaggregation

the existence of asymptotically unbiased nonnegative quadratic estimates of variance components in anova models abstract conditions for the existence of asymptotically unbiased nonnegative quadratic estimates of individual variance components in general anova models are investigated a simple necessary condition for general models is obtained necessary and sufficient conditions are obtained in the case of random balanced nested classification models and the random unbalanced one way model the investigation demonstrates the need for exercising caution when employing nonnegative quadratic estimates of individual variance components in practice

approximation by piecewise regular maps abstract a real algebraic variety w of dimension m is said to be uniformly rational if each of its points has a zariski open neighborhood which is biregularly isomorphic to a zariski open subset of r m let l be any nonnegative integer we prove that every map of class c l from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the c l topology by piecewise regular maps of class c k where k is an arbitrary integer satisfying k l next we derive consequences regarding algebraization of topological vector bundles

carleman approximation by conformal minimal immersions and directed holomorphic curves abstract let r be an open riemann surface in this paper we prove that every continuous function m r n n 3 defined on a divergent jordan arc m r can be approximated in the carleman sense by conformal minimal immersions thus providing a new generalisation of carleman s theorem in fact we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property furthermore the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map as a consequence we give an approximate solution to a plateau problem for divergent jordan curves in the euclidean spaces

closed holomorphic 1 forms without zeros on stein manifolds we prove that a stein mainfold admits a closed holomorphic 1 form without zeros in every class of the first cohomology group we also prove an approximation result for closed holomorphic 1 forms without zeros defined in a neighborhood of a compact subset

embedded minimal surfaces in mathbb r n in this paper we prove that every conformal minimal immersion of an open riemann surface into mathbb r n for n ge 5 can be approximated uniformly on compacts by conformal minimal embeddings see theorem 1 1 furthermore we show that every open riemann surface carries a proper conformal minimal embedding into mathbb r 5 see theorem 1 2 one of our main tools is a mergelyan approximation theorem for conformal minimal immersions to mathbb r n for any n ge 3 which is also proved in the paper see theorem 5 3

spectra of weighted composition operators with automorphic symbols abstract let be an automorphism of the open unit disc d for such we investigate the spectra of invertible weighted composition operators u c acting on a wide class of analytic function spaces this class contains for example hardy spaces h p d weighted bergman spaces a p d and weighted banach spaces of h type we present new techniques for deducing the spectrum and for calculating the spectral radius of u c we also characterize the fredholmness of weighted composition operators on h p d and a p d

pluricapacity and approximation numbers of composition operators abstract for suitable bounded hyperconvex sets in c n in particular the ball or the polydisk we give estimates for the approximation numbers of composition operators c h 2 h 2 when is relatively compact in involving the monge ampere capacity of

compact composition operators on bergman orlicz spaces we construct an analytic self map of the unit disk and an orlicz functionfor which the composition operator of symbol is compact on the hardy orlicz space h but not on the bergman orlicz space b for that we first prove a carleson embedding theorem and then characterize the compact ness of composition operators on bergman orlicz spaces in terms of carleson function of order 2 we show that this carleson function is equivalent to the nevanlinna counting function of order 2 mathematics subject classification primary 47b33 secondary 30d50 30d55 46e15

covariance hypotheses which are linear in both the covariance and the inverse covariance both the covariance and the inverse covariance are products of models each of which consists of either i independent identically distributed random vectors which have a covariance with a real complex or quaternion structure or ii independent identically distributed random vectors with a parametrization of the covariance which is given by means of the clifford algebra the models i are well known for models ii we have found under the assumption that the distribution is nornal the exact distributions of the maximum likelihood estimates and the likelihood ratio test statistics 1 introduction the purpose of the present paper is to describe for the family of normal distributions the structure of those statistical hypotheses models which are linear in both the covariance and the inverse covariance in section 2 we show that under such a hypothesis the problem of maximum likelihood estimation of the covariance has an explicit solution and that these hypotheses are parametrized by jordan algebras quadratic subspaces based on the theory of jordan algebras it is then possible to give a canonical form for such hypotheses in section 3 we show that the jordan algebras are products of so called simple jordan algebras and in section 4 we prove that the hypotheses can be decomposed into products of hypotheses parametrized by simple jordan algebras the simple jordan algebras are classified according to their so called degree the only simple jordan algebras of degree 1 is the set of real numbers and corresponding to that we have the hypothesis under which the covariance is proportional to a known covariance in section 5 we give a canonical form for the hypotheses which are parametrized by simple jordan algebras of degree greater than 3 it tums out that these hypotheses are equivalent to those consisting of independent identically distributed random vectors which have a covariance with a real complex or quatemion structure and the distributions of the maximum likelihood estimates are given by wishart 1928 goodman 1963 and andersson 1975 respectively the statistical hypotheses which are parametrized by simple jordan algebras of degree 2 are new and in section 6 we give a complete solution to the problems of maximum likelihood inference it turns out that the distribution problems are not more complicated than those for the two dimensional normal distribution the mathematical structure of the hypotheses is closely related to the theory of spinors in quantum mechanics based on that theory we give a canonical form for such hypotheses in section 7

criteria for the density property of complex manifolds in this paper we suggest new effective criteria for the density property this enables us to give a trivial proof of the original anders en lempert result and to establish almost free of charge the algebraic density property for all linear algebraic groups whose connected components are different from tori or c as another application of this approach we tackle the question asked among others by f forstneri v c about the density of algebraic vector fields on euclidean space vanishing on a codimension 2 subvariety

extending holomorphic sections from complex subvarieties in this paper we treat the classical problem of extending holomorphic mappings and sections from closed complex subvarieties of a complexmanifold our main results theorems 1 1 and 1 4 extend those of grauert gr2 and cartan car results of this type are commonly referred to as the okagrauert principle on stein manifolds our methods are similar to those developed in the papers of gromov gro henkin and leiterer hl2 and the authors fp1 fp2 the main addition here is the interpolation of a given holomorphic section on a complex subvariety of a stein manifold our globalization scheme follows very closely the one developed in fp2 to state our first result we recall the notion of a dominating spray introduced by m gromov gro sect 0 5 given a holomorphic vector bundle p e y over a complex manifold y we denote by 0y ey p 1 y the zero element in the fiberey and we observe thatey is ac linear subspace of the tangent space t0ye definition 1 a spray on a complex manifold y is a holomorphic vector bundle p e y together with a holomorphic map s e y such that for each y y s 0y y and the derivative ds t0ye tyy maps ey surjectively onto tyy

a homotopy theorem for oka theory we prove a homotopy theorem for sheaves its application shortens and simplifies the proof of many oka principles such as gromov s oka principle for elliptic submersions

an equivariant version of grauert s oka principle for example a theorem of oka asserts that every holomorphic line bundle over a domain of holomorphy in 12 is holomorphicaily trivial if and only if it is topologically trivial in g3 grauer t proves that the category of bundles over a stein space x with a complex lie group as a structure group is the same up to isomorphism if considered in the topological or complex analytic category we would like to underl ine that grauert s method of proof also has far reaching consequences see e g fr1 fr2 in this article we prove the equivariant version of grauert s oka principle for a compact lie group of holomorphic t ransformations on x the precise statements can be found in w w and w for example as a consequence of the main results we obtain

holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space we construct holomorphic families of proper holomorphic embeddings of mathbb c k into mathbb c n 0 textless k textless n 1 so that for any two different parameters in the family no holomorphic automorphism of mathbb c n can map the image of the corresponding two embeddings onto each other as an application to the study of the group of holomorphic automorphisms of mathbb c n we derive the existence of families of holomorphic mathbb c actions on mathbb c n n ge5 so that different actions in the family are not conjugate this result is surprising in view of the long standing holomorphic linearization problem which in particular asked whether there would be more than one conjugacy class of mathbb c actions on mathbb c n with prescribed linear part at a fixed point

homotopy principles for equivariant isomorphisms let g be a reductive complex lie group acting holomorphically on stein manifolds x and y let px x qx and py y qy be the quotient mappings when is there an equivariant biholomorphism of x and y a necessary condition is that the categorical quotients qx and qy are biholomorphic and that the biholomorphism sends the luna strata of qx isomorphically onto the corresponding luna strata of qy fix we demonstrate two homotopy principles in this situation the first result says that if there is a g diffeomorphism x y inducing which is g biholomorphic on the reduced fibres of the quotient mappings then is homotopic through g diffeomorphisms satisfying the same conditions to a g equivariant biholomorphism from x to y the second result roughly says that if we have a g homeomorphism x y which induces a continuous family of gequivariant biholomorphisms of the fibres px 1 q and py 1 q for q qx and if x satisfies an auxiliary property which holds for most x then is homotopic through g homeomorphisms satisfying the same conditions to a g equivariant biholomorphism from x to y our results improve upon those of our earlier paper j reine angew math 706 2015 193 214 and use new ideas and techniques

jordan algebras and bayesian quadratic estimation of variance components the authors consider a linear model ey x cov y v where v is a quadratic subspace a jordan algebra using this property it is shown how multiplication computation of g inverses determination of a neutral element can be done more efficiently in v finally this technique is applied to obtain formulae for locally optimal estimators and bayesian estimators

some examples of composition operators and their approximation numbers on the hardy space of the bi disk we give examples of composition operators c phi on h 2 d 2 showing that the condition phi infty 1 is not sufficient for their approximation numbers a n c phi to satisfy lim n to infty a n c phi 1 sqrt n 1 contrary to the 1 dimensional case we also give a situation where this implication holds we make a link with the monge amp ere capacity of the image of phi

stein structures and holomorphic mappings we prove that every continuous map from a stein manifold x to a complex manifold y can be made holomorphic by a homotopic deformation of both the map and the stein structure on x in the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank the analogous result holds for any compact hausdorff family of maps but it fails in general for a noncompact family our main results are actually proved for smooth almost complex source manifolds x j with the correct handlebody structure the paper contains another proof of eliashberg s int j math 1 29 46 1990 homotopy characterization of stein manifolds and a slightly different explanation of the construction of exotic stein surfaces due to gompf ann math 148 2 619 693 1998 j symplectic geom 3 565 587 2005

proof of a conjecture of zahariuta concerning a problem of kolmogorov on the e entropy we prove a conjecture of zahariuta which itself solves a problem of kolmogorov on the e entropy of some classes of analytic functions for a given holomorphically convex compact subset k in a pseudoconvex domain d in c n zahariuta s conjecture consists in approximating the relative extremal function u k d uniformly on any compact subset of d k by pluricomplex green functions on d with logarithmic poles in the compact subset k

density property for hypersurfaces uv p bar x we study hypersurfaces of mathbb c n 2 bar x u v given by equations of form uv p bar x where the zero locus of a polynomial p is smooth reduced the main result says that the lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the lie algebra of all algebraic vector fields consequences of this result for some conjectures of affine algebraic geometry and for the oka grauert gromov principle are discussed

an interpolation theorem for proper holomorphic embeddings given a stein manifold x of dimension n 1 a discrete sequence a j subset x and a discrete sequence b j subset mathbb c m where m ge n left frac 3n 2 right 1 there exists a proper holomorphic embedding f colon x hookrightarrow mathbb c m satisfying f a j b j for every j 1 2

an equivariant parametric oka principle for bundles of homogeneous spaces we prove a parametric oka principle for equivariant sections of a holomorphic fibre bundle e with a structure group bundle mathscr g on a reduced stein space x such that the fibre of e is a homogeneous space of the fibre of mathscr g with the complexification k mathbb c of a compact real lie group k acting on x mathscr g and e our main result is that the inclusion of the space of k mathbb c hbox equivariant holomorphic sections of e over x into the space of k hbox equivariant continuous sections is a weak homotopy equivalence the result has a wide scope we describe several diverse special cases we use the result to strengthen heinzner and kutzschebauch s classification of equivariant principal bundles and to strengthen an oka principle for equivariant isomorphisms proved by us in a previous paper

oka s principle for holomorphic sections of elliptic bundles 2 the h principle for locally trivial fibrations 2 1 the h principle over small subsets 2 2 the h principle over totally real submanifolds 2 3 local extension of the h principle 2 4 localizable extensions 2 5 the h principle for cm bundles 2 6 manifolds with totally real souls 2 7 totally real extensions 2 8 nicely localizable extensions 2 9 the basic h principle for z x 2 10 homomorphisms and holomorphic maps of rank r

a spectral radius type formula for approximation numbers of composition operators abstract for approximation numbers a n c of composition operators c on weighted analytic hilbert spaces including the hardy bergman and dirichlet cases with symbol of uniform norm lim n a n c 1 n e 1 cap d where cap d is the green capacity of d in d this formula holds also for h p with 1 p

oka properties of ball complements let n 1 be an integer we prove that holomorphic maps from stein manifolds x of dimension n to the complement mathbb c n setminus l of a compact convex set l subset mathbb c n satisfy the basic oka property with approximation and interpolation if l is polynomially convex then the same holds when 2 dim x le n we also construct proper holomorphic maps immersions and embeddings x rightarrow mathbb c n with additional control of the range thereby extending classical results of remmert bishop and narasimhan

linear toeplitz covariance structure models with optimal estimators of variance components this paper deals with the problem of optimal quadratic unbiased estimation for statistical models with linear toeplitz covariance structure the main result is a necessary and sufficient condition for these models to have an optimal unbiased estimator for any linear combination of variance components this result is obtained by means of special jordan algebras which are a powerful tool to characterize optimality in quadratic unbiased estimation

compact composition operators on the dirichlet space and capacity of sets of contact points we prove several results about composition operators on the dirichlet space d for every compact set k d of logarithmic capacity capk 0 there exists a schur function both in the disk algebra a d and in d such that the composition operator c is in all schatten classes sp d p 0 and for which k eit eit 1 eit eit 1 for every bounded composition operator c on d and every d the logarithmic capacity of eit eit is 0 every compact composition operator c on d is compact on b 2 and on h 2 in particular c is in every schatten class sp p 0 both on h2 and on b2 there exists a schur function such that c is compact on h 2 but which is not even bounded on d there exists a schur function such that c is compact on d but in no schatten class sp d

holomorphic flexibility properties of complex manifolds we obtain results on approximation of holomorphic maps by algebraic maps the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds and the homotopy principle for holomorphic submersions of stein manifolds to certain algebraic manifolds

embedding some riemann surfaces into mathbb c 2 with interpolation we prove that several types of open riemann surfaces including the finitely connected planar domains embed properly into mathbb c 2 such that the values on any given discrete sequence can be arbitrarily prescribed

flexible varieties and automorphism groups given an irreducible affine algebraic variety x of dimension n 2 we let saut x denote the special automorphism group of x that is the subgroup of the full automorphism group aut x generated by all one parameter unipotent subgroups we show that if saut x is transitive on the smooth locus x reg then it is infinitely transitive on x reg in turn the transitivity is equivalent to the flexibility of x the latter means that for every smooth point x x reg the tangent space t x x is spanned by the velocity vectors at x of one parameter unipotent subgroups of aut x we also provide various modifications and applications

oka s principle for holomorphic submersions with sprays the oka principle is a fundamental principle of complex analysis which says that on stein manifolds closed complex submanifolds of affine spaces analytic problems of cohomological or even homotopical nature have only topological obstructions the story began in 1939 when k oka proved that on a domain of holomorphy a secondcousin problem is solvable by holomorphic functions if it is solvable by continuous functions oka oka s result has the following equivalent formulation ife x is a holomorphicc bundle over adomainof holomorphy then every continuous section of e is homotopic to a holomorphic section in 1957 h grauert gra extended oka s result to sections of holomorphic fiber bundles with complex homogeneous fibers over stein manifolds see also the papers car fr hl in 1989 m gromov gr1 announced a major generalization of grauert s theorem replacing complex lie groups and homogeneous spaces by a larger class of complex manifolds with a spray a spray on a complex manifold y is a holomorphic vector bundle p e y with a holomorphic map s e y the spray map which is the identity on the zero section and whose restriction to each fibre of e is a submersion at zero such sprays exist on complex homogeneous spaces on complements of affine algebraic subvarieties of codimension at least two and more generally on manifolds whose holomorphic tangent bundle is spanned by finitely many c complete holomorphic vector fields the notion of spray extends in a natural way to submersions the following is theorem 4 5 in gr1 if h z x is a holomorphic submersion onto a stein manifold x such that every point x x has an open neighborhood u x such that z u h 1 u u admits a fiber dominating spray then the inclusion of the

the oka grauert principle without induction over the base dimension we give a new proof of grauert s theorem on oka s principle gra1 gra2 gra3 see also c in the case of smooth stein manifolds which does not use induction over the base dimension instead we use induction over the levels of a strictly plurisubharmonic exhausting function grauert s bump method the present paper is an edited version of our preprint h l2 from 1986 which is difficult to find and of bad printing quality we did not publish the paper in a journal at that time because we planned to write a book containing it but the book has not been written until now on the other hand in the meantime some interest to this proof appeared see e g the work of gromov gro therefore we think a publication of our proof could be useful even with a delay of 10 years this proof gives also some oka principle for arbitrary pseudoconvex manifolds see theorem 1 3 below for vector bundles vector bundles without further mention will refer to topological vector bundles with fibre cr this oka principle contains the following