a note on maqusi s proofs and truncation error bounds in the dyadic walsh sampling theorem in 1 and 2 bounds for the truncation error are presented which arises when a sequency limited signal function is reconstructed by finite walsh sampling sums the knowledge of the properties that sequency limited functions have leads to an immediate proof of the dyadic sampling theorem and truncation turns out to be a trivial affair

tensor product of n isometries abstract a hilbert space operator a is an m isometry if k 0 m 1 k m k a m k a m k 0 we prove that the tensor product of an m isometry a with an n isometry b is an m n 1 isometry this proves a conjecture of botelho and jamison 5 remark 5 1

bounded analytic functions in the dirichlet space in this paper we study the hilbert space of analytic functions with finite dirichlet integral in a connected open set c2 in the complex plane we show that every such function can be represented as a quotient of two bounded analytic functions each of which has a finite dirichlet integral this has several consequences for the structure of invariant subspaces of the algebra of multiplication operators on the dirichlet space in case f2 is simply connected namely we show that every nontrivial invariant subspace contains a nontrivial bounded function that each two nontrivial invariant subspaces have a nontrivial intersection that is the algebra is cellular indecomposable and that each nontrivial invariant subspace has the codimension one property with respect to certain multiplication operators

dirichlet type spaces on the unit ball and joint 2 isometries abstract we obtain a formula that relates the spherical moments of the multiplication tuple on a dirichlet type space to a complex moment problem in several variables this can be seen as the ball analogue of a formula originally invented by richter in 23 we capitalize on this formula to study dirichlet type spaces on the unit ball and joint 2 isometries

invariant subspaces of the dirichlet shift and pseudocontinuations in this paper we study extremal functions for invariant subspaces x of the dirichlet shift i e solutions v of the extremal problem sup if n 0 1 1f lid f e x f o here n is the smallest nonnegative integer such that the sup is positive it is known that such a function v generates x we show that the derivative z o has a pseudocontinuation to the exterior disc this pseudocontinuation is an analytic continuation exactly near those points of the unit circle where 0 is bounded away from zero we also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace

unbounded symmetrizable idempotents abstract the relationship between closed unbounded idempotents and dense decompositions of a hilbert space is explored by extending the notion of compatibility between closed subspaces and positive bounded operators

closed derivations inc algebras the early study of derivations on operator algebras and banach algebras concerned the continuity of them silov 21 showed that there is no norm on f 0 1 under which 0 1 becomes a banach algebra in connection with this result kaplansky posed a question whether every derivation on a semi simple banach algebra is continuous johnson and sinclair 11 showed that the answer is affirmative see 13 19 22 in sect 2 we shall give a generalization of the silov theorem which is stated as follows let fi be a closed derivation in a c algebra 9 i suppose that is dense in 9 i i f there exists a norm on under which becomes n l a banach algebra then 6 is bounded on 9 i as a corollary we shall show that a closed derivation in a c algebra is bounded if the range is contained in the domain

a representation theorem for cyclic analytic two isometries a bounded linear operator t on a complex separable hilbert space z is called a 2 isometry if t 2t2 2t t i o we say that t is analytic if nn 0 tnt 0 in this paper we show that every cyclic analytic 2 isometry can be represented as multiplication by z on a dirichlet type space d 8 here ze denotes a finite positive borel measure on the unit circle for two measures u and v the 2 isometries obtained as multiplication by z on d 8 and d v are unitarily equivalent if and only if u v we also investigate similarity and quasisimilarity of these 2 isometries and we apply our results to the invariant subspaces of the dirichlet shift

cyclicity and invariant subspaces in dirichlet spaces let be a positive finite measure on the unit circle and d the associated dirichlet space the generalized brown shields conjecture asserts that an outer function f d is cyclic if and only if c z f 0 where c is the capacity associated with d and z f is the zero set of f in this paper we prove that this conjecture is true for measures with countable support we also give in this case a complete and explicit characterization of invariant subspaces

reducing subspaces of non analytic toeplitz operators on weighted hardy and dirichlet spaces of the bidisk abstract we characterize reducing subspaces of a class of non analytic toeplitz operators on hilbert spaces of holomorphic functions on the bidisk including the hardy space and the dirichlet space of the bidisk these results compliment a recent characterization of the reducing subspaces for this class of non analytic toeplitz operators on the bergman space of the bidisk

a class of asymptotic tests for principal component vectors in this paper the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered a class of invariant asymptotic tests based on the sample covariance matrix is derived tests in this class are shown to be consistent and their local power functions are given the arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of the sample covariance matrix the results are shown to generalize when the procedures are based on any affine invariant m estimate of scatter and when the population is elliptical

aspects of nonnormality for iterative methods abstract recently new optimal krylov subspace methods have been discovered for normal matrices in light of this novel ways to quantify nonnormality are considered in connection with various families of matrices we use as a criterion how for a given matrix these iterative methods introduced can be employed via e g inexpensive matrix factorizations the unitary orbit of the set of binormal matrices provides a natural extension of normal matrices its elements yield polynomially normal matrices of moderate degree in this context several matrix nearness problems arise

maximal minimal and primary invariant subspaces abstract let x be a complex infinite dimensional banach space an operator l on x is called of subcritical class if n 1 n 3 2 log l n t is an operator on x whose iterates have norms of polynomial growth we prove that if t has a range of finite codimension and a left inverse of subcritical class then every maximal invariant subspace of t has codimension one and if t has a finite dimensional kernel and a right inverse of subcritical class then every minimal invariant subspace of t is one dimensional using these results we obtain new information on the invariant subspace lattices of shifts and backward shifts on a wide class of banach spaces of analytic functions on the unit disc we also introduce the notion of primary invariant subspaces and determine their structure for a large class of shifts

kernel and capacity estimates in dirichlet spaces abstract we study the capacity in the sense of beurling deny associated with the dirichlet space d where is a finite positive borel measure on the unit circle first we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of d it allows us to give an estimate of the capacity of points and arcs of the unit circle we also provide new conditions on closed sets to be polar our method is based on sharp estimates of norms of some outer functions which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted sobolev space

m isometric operators and their local properties abstract in this paper we give necessary and sufficient conditions for a bounded linear hilbert space operator to be an m isometry for an unspecified m written in terms of conditions that are applied to one vector at a time we provide criteria for orthogonality of generalized eigenvectors of an a priori unbounded linear operator t on an inner product space that correspond to distinct eigenvalues of modulus 1 we also discuss a similar question of when jordan blocks of t corresponding to distinct eigenvalues of modulus 1 are orthogonal

estimates for n widths of multiplier operators of multiple walsh series abstract estimates for kolmogorov and gelfand n widths of multiplier operators of multiple walsh series are obtained upper and lower bounds are established for n widths of general multiplier operators these results are applied to get upper and lower bounds for n widths of specific multiplier operators which generate sets of finitely and infinitely differentiable functions in the dyadic sense it is shown that these estimates have order sharp in various important cases

on higher order selfadjoint operators abstract for a positive integer m and a hilbert space h an operator t in b h the space of all bounded linear operators on h is called m selfadjoint if k 0 m 1 k m k t k t m k 0 in this paper we show that if t b h and the spectrum of t consists of a finite number of points then it is m selfadjoint if and only if it is an n jordan operator for some integer n moreover we prove that if t is m selfadjoint then t is nilpotent when it is quasinilpotent then we characterize m selfadjoint weighted shift operators also we show that if t is m selfadjoint then so is p t when p z is a polynomial with real coefficients after that we investigate an elementary operator and a generalized derivation operator on the hilbert schmidt class c 2 h which are m selfadjoint finally we prove that no m selfadjoint operator on an infinite dimensional hilbert space can be n supercyclic for any n 1

how many variables should be entered in a regression equation abstract the optimal number of regressors is determined to minimize mean squared prediction error and is shown to be a small fraction of the number of data points as the number of regressors grows large the sp criterion provides an asymptotically optimal rule for the number of variables to enter

decomposing algebraic m isometric tuples abstract we show that any m isometric tuple of commuting algebraic operators on a hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables

which de branges rovnyak spaces are dirichlet spaces and vice versa abstract we investigate for which pairs b the de branges rovnyak space h b is equal to the generalized dirichlet space d

hermite and laguerre polynomials with complex matrix arguments this paper defines and discusses the complex hermite and laguerre polynomials associated with the complex matrix variate normal and wishart distributions respectively various properties of these polynomials are investigated including generating functions rodrigues formulae differential and integral versions and series expressions these polynomials are also discussed from the viewpoint of the multivariate complex meixner distributions we present applications in asymptotic distribution theory on the complex stiefel manifold the theory of complex zonal polynomials is of great use in the derivations

real linear kronecker product operations the kronecker product in the real linear matrix analytic setting is studied more versatile operations are proposed such generalizations are of interest for the same reasons the standard kronecker product is to give an example new preconditioning ideas are suggested in connection with this several formulae for the inverse are devised orthogonal decompositions of real entried matrices are derived through introducing new kronecker product svds matrix equations are given to illustrate how the kronecker product structures introduced can arise

a solution to the cauchy dual subnormality problem for 2 isometries abstract the cauchy dual subnormality problem asks whether the cauchy dual operator t t t t 1 of a 2 isometry t is subnormal in the present paper we show that the problem has a negative solution the first counterexample depends heavily on a reconstruction theorem stating that if t is a 2 isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition then t is subnormal if and only if t satisfies the unperturbed kernel condition the second counterexample arises from a 2 isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the cauchy dual subnormality problem in this context we prove that if t is a 2 isometry satisfying the kernel condition or a quasi brownian isometry then t is subnormal we construct a 2 isometric adjacency operator t of a rooted directed tree such that t does not satisfy the kernel condition t is not a quasi brownian isometry and t is subnormal

edgeworth and saddle point approximations for random rectangular matrices abstract we present methods for obtaining asymptotic expansions for the probability density functions of sample means of independent and identically distributed random rectangular matrices the expansions are of the edgeworth saddle point and generalized edgeworth types an example of exact saddle point approximations and applications of the methods are illustrated

strict isometries of arbitrary orders abstract we consider the elementary operator l acting on the hilbert schmidt class c 2 h given by l t atb with a and b bounded operators on a separable hilbert space h in this paper we establish results relating isometric properties of l with those of the defining symbols a and b we also show that if a is a strict n isometry on a hilbert space h then i a a a 2 a 2 a n 1 a n 1 is a linearly independent set of operators this result allows to extend further the isometric interdependence of l and its symbols in particular we show that if l is a p isometry then a is a strict p 1 or p 2 isometry if and only if b is a strict 2 or 3 isometry

the minus order and range additivity abstract we study the minus order on the algebra of bounded linear operators on a hilbert space by giving a characterization in terms of range additivity we show that the intrinsic nature of the minus order is algebraic applications to generalized inverses of the sum of two operators to systems of operator equations and to optimization problems are also presented

which de branges rovnyak spaces have complete nevanlinna pick property abstract we characterize the de branges rovnyak spaces with complete nevanlinna pick property our method relies on the general theory of reproducing kernel hilbert spaces

elementary properties of isometries on a hilbert space inspired by recent works on m isometries for a positive integer m in this paper we introduce the classes of isometries and unitaries on a hilbert space we show that an isometry on a finite dimensional complex hilbert space h with dimension n is in fact an 2n 1 isometry we describe the spectra of such operators study the quasinilpotent perturbations of isometries and characterize when tensor products of isometries are also isometries as a surprising by product we obtain a generalization of nagy foias langer decomposition of a contraction into an unitary and a completely nonunitary contraction

generalized noncentral hermite and laguerre polynomials in multiple matrices abstract we are concerned with the generalized noncentral hermite and laguerre polynomials in multiple matrices we define the generalized hermite polynomials h mn k r o y q a r in q m n rectangular matrix arguments y1 yq y q and r m m constant matrices a1 ar a r q r associated with the joint distribution of q independent m n rectangular matrix variate standard normal distributions and present various properties of theirs the discussion of the generalized hermite polynomials h m k r o x q a r in q m m symmetric matrix arguments x1 xq x q associated with the joint distribution of q independent m m symmetric matrix variate normal distributions introduced by chikuse is further facilitated the generalized noncentral hermite polynomials are defined as h mn k r o y n q a r and h m k r o x m q a r for q m n rectangular and m m symmetric matrices n q and m q respectively extending results of chikuse and davis to the noncentral case we define and discuss the generalized noncentral laguerre polynomials l u q k r o x q a r q in q m m noncentrality matrices 1 q q associated with the joint distribution of q independent noncentral wishart distributions many unsolved problems can now be resolved by introducing these noncentral polynomials in multiple matrices specifically both the differential and integral versions of the generalized multivariate rodrigues formulae are obtained for all of these polynomials they are of great use in multivariate distribution theory a simple example of applications is presented

a m isometries on hilbert spaces abstract a bounded linear operator t on a hilbert space h is called an a m isometry for some positive operator a on h and integer m if k 0 m 1 m k m k t k a t k 0 we give some properties of a m isometries in particular we focus on spectral properties and the relation between a m isometries and m isometries also we obtain some dynamic properties of a m isometries as a negative answer to 22 question 1 with an example of an a isometric which is n supercyclic and sufficient conditions for an a m isometry to be not n supercyclic moreover we prove that the perturbation of a m isometry by a bigger class than nilpotent operators is not n supercyclic

spherical functions on the grassmann manifold and generalized jacobi polynomials part 1 abstract generalized jacobi polynomials constitute a complete system of orthogonal symmetric polynomials with respect to a multivariate beta measure this paper gives an explicit formula for these polynomials in terms of the author s invariant polynomials in two matrix arguments and derives the orthogonality property explicit formulas are also given for certain constants which appear in the theory of zonal polynomials in cases where the partitional parameters have not more than two nonzero parts zonal spherical functions on the grassmann manifold may be expressed in terms of generalized jacobi polynomials in part 2 of this paper it will be shown that these zonal spherical functions have an integral representation which generalizes the well known integrals for legendre and gegenbauer polynomials of even order in particular cases this integral representation may be used to construct the zonal and associated spherical functions in terms of univariate special functions

density estimation on the spaces of symmetric and rectangular matrices abstract this paper develops the theory of density estimation on the space s m of all m m symmetric matrices and on the space r m p of all m p rectangular matrices we consider the two methods kernel density estimation and density estimation by orthogonal series and illustrate them with examples for the normal kernel density functions and for the hermite orthonormal bases with symmetric and rectangular matrix arguments on s m and r m p respectively the problem of estimating unknown joint density functions of multiple random matrices is also discussed

a biharmonic maximum principle for hyperbolic surfaces let w be a simply connected two dimensional riemannian manifold with a cysmooth metric ds we can always introduce local isothermal coordinates near any point of w that is such coordinates x y that the metric is represented in the form ds x y 1 4 o x y dx dy for some positive weight function o and since w is orientable as a two dimensional simply connected manifold these local isothermal coordinates can serve as conformal charts for a complex structure on w see 1 pp 124 126 the k be uniformization theorem then says that w is conformally equivalent to one of the three sets the riemann sphere s 1 4 cw fyg the complex plane c or the open unit disk d this equivalence together with the choice of isothermal coordinates allows us to identify w with one of the above three sets w supplied with the isothermal riemannian metric

unbounded nilpotents and idempotents abstract this paper is concerned with unbounded operators in hilbert space with domain containing their range and in particular investigates unbounded nilpotents and idempotents

an isometry plus a nilpotent operator is an m isometry applications abstract we prove that if an isometry a and a nilpotent operator q of order n commute then a q is a strict 2 n 1 isometry as an application of the main result we prove that a q cannot be n supercyclic for any n also we find an m isometry with prescribed spectrum k where k is the closed unit disk or a compact subset of the unit circle

dyadic calculus and sampling theorems for functions with multidimensional domain i general theory the theory of dyadic walsh analysis for functions of several variables is presented to give a rather complete presentation of dyadic analysis a multidimensional version of the dyadic derivative introduced by gibbs and butzer wagner is developed moreover using walsh fourier transform techniques connections between dyadic differentiability and smoothness conditions are established

complete nevanlinna pick property of dirichlet type spaces we prove that all dirichlet type spaces of functions analytic in the unit disk whose derivatives are square area integrable with superharmonic weights have complete nevanlinna pick reproducing kernels as a corollary we obtain a commutant lifting theorem for cyclic analytic two isometries c 2002 elsevier science usa

classification of hereditary matrices abstract a classical approach used to obtain basic facts in the theory of square matrices involves an analysis of the relationship between polynomials p in one variable and square matrices t such that p t 0 we consider matrices and operators which satisfy a different type of polynomial constraint let h be a complex hilbert space t be a bounded linear transformation of h t be the adjoint of t and c x y be the algebra of polynomials in x and y with complex coefficients for a polynomial p c x y in two variables with complex coefficients define p t m n 0 p m n t n t m where p m n is the coefficient of ynxm in the expansion of p in a power series about the point 0 0 t is called a root of p if and only if p t 0 note that if p c x y is a polynomial in the single variable x then the definition of p t given here agrees with the classical definition in this paper we study the relationships which p t 0 forces between p and t when t is an algebraic operator i e there exists n 1 and complex numbers a0 an 1 such that 0 a0 a1t an 1tn 1 tn the classification starts with the following observation suppose p c x y and an algebraic operator t l h satisfy p t 0 then certain subspaces of h which are invariant for t must be orthogonal or certain coefficients of p must vanish this leads to the notions of a graph attached to each p c x y and a graph attached to each square matrix t for diagonalizable t a necessary and sufficient graph theoretic condition for solving p t 0 is given for nondiagonalizable t this condition is necessary but not sufficient the use of these graphs does however reduce the problem to the problem of solving the equation p t 0 for t with exactly one or two eigenvalues for t with one eigenvalue we give a necessary and sufficient condition for solving p t 0 this leaves the case of solving p t 0 when t has exactly two eigenvalues this problem mixes algebra involving polynomials with matrix theory we show that it is equivalent to the purely algebraic problem of determining if equations of the form i j eci jxi r j s 0 have solutions of finite support with certain nonvanishing properties we call these equations bi hankel equations subordinate to a given subset e of the lattice of integer pairs i j 0 i m 1 0 j n 1 it turns out that there is an algorithm which uses grobner bases for determining if the type of solution we seek exists and for computing it

algebraic properties of operator roots of polynomials abstract properties of m selfadjoint and m isometric operators have been investigated by several researchers particularly interesting to us are algebraic properties of nilpotent perturbations of such operators mccullough and rodman showed in the nineties that if q n 0 and a is a selfadjoint operator commuting with q then the sum a q is a 2 n 1 selfadjoint operator very recently bermudez martinon and noda proved a similar result for nilpotent perturbations of isometries via a new approach we obtain simple proofs of these results and other generalizations to operator roots of polynomials

an extension of the dyadic calculus with fractional order derivatives general theory a new dyadic calculus is set up principally based upon that introduced by butzer and wagner 1972 1975 and zelin he 1983 the extended dyadic derivative of a function which is formulated for fractional orders is roughly the euler summation process applied to the fourier walsh series off after it has first been equipped with a certain multiplicative factor this extended calculus is not only applicable to piecewise constant functions as is the classical dyadic derivative but also to piecewise polynomials this paper presents the full general theory of this extended dyadic calculus introduction and justification of the dyadic derivative its fundamental properties and those of its eigenvalues the corresponding anti differentiation operator a type of counterpart of the fundamental theorem of the newton leibniz calculus in the frame of walsh analysis applications and further theory are dealt with in the second paper in the series

some results on higher order isometries and symmetries products and sums with a nilpotent operator abstract we study the sum of an m isometry or an m symmetric operator with a nilpotent operator we also study the product of two m isometries or two n symmetries we obtain several theorems generalizing previous work our method is straightforward and thus offers direct and simple insights into why these theorems are true

common principal components in k groups abstract this article generalizes the method of principal components to so called common principal components as follows consider the hypothesis that the covariance matrices i for k populations are simultaneously diagonalizable that is there is an orthogonal matrix such that i is diagonal for i 1 k i derive the normal theory maximum likelihood estimates of the common component i matrices and the log likelihood ratio statistics for testing this hypothesis the solution has some favorable properties that do not depend on normality assumptions numerical examples illustrate the method applications to data reduction multiple regression and nonlinear discriminant analysis are sketched

walsh spectral analysis in this paper we review recent work on the stochastic approach of walsh spectral analysis we are mainly interested in the properties of the finite walsh transform which in turn allow us to derive properties of the estimators of the walsh spectrum we further discuss linear dyadic filters and walsh characteristic functions a rather complete list of references is given

elementary operators which are m isometries abstract the study of m isometry elementary operator of length one is initiated by botelho and jamison several authors have contributed to obtain this result if the two operator coefficients involved are m isometries then the elementary operator is m isometry on the hilbert schmidt operator ideal in this paper we show that the converse of this result is also true we also give a necessary and sufficient condition for the generalized derivation to be an m isometry the sufficiency follows from results on the sum of m isometries and nilpotent operators in two recent papers

products of m isometries abstract an operator t on a banach space x is called an m p isometry if it satisfies the equality k 0 m m k 1 m k t k x p 0 for all x x in this paper we prove that if t is an n p isometry s is an m p isometry and they commute then ts is an m n 1 p isometry this result applied to elementary operators of length 1 defined on the hilbert schmidt class proves a conjecture in 11

invariant subspaces in banach spaces of analytic functions we study the invariant subspace structure of the operator of multiplication by z mz on a class of banach spaces of analytic functions for operators on hilbert spaces our class coincides with the adjoints of the operators in the cowendouglas class 1 q we say that an invariant subspace x satisfies cod x 1 if zx has codimension one in x we give various conditions on invariant subspaces which imply that codx 1 in particular we give a necessary and sufficient condition on two invariant subspaces x x with cod x cod x 1 so that their span again satisfies cod x v x 1 this result will be used to show that any invariant subspace of the bergman space l p 1 which is generated by functions in l2p must satisfy cod x 1 for an invariant subspace x we then consider the operator s m 1xl under some extra assumption on the domain of holomorphy we show that the spectrum of s coincides with the approximate point spectrum iff cod x 1 finally in the last section we obtain a structure theorem for invariant subspaces with cod x 1 this theorem applies to dirichlet type spaces

elementary operators and the aluthge transform abstract we characterize essential normality for certain elementary operators acting on the hilbert schmidt class we find the aluthge transform of an elementary operator of length one we show that the aluthge transform of an elementary 2 isometry need not be a 2 isometry we also characterize hermitian elementary operators of length 2

asymptotic inference for eigenvectors abstract asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors these asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest applications of the general results to principal components analysis and canonical variate analysis are given author

de branges rovnyak spaces and dirichlet spaces sarason has shown that the local dirichlet spaces d may be considered as manifestations of de branges rovnyak spaces h b and has used this identification to give a new proof that the spaces d are star shaped we investigate which other dirichlet spaces d arise as de branges rovnyak spaces and which other de branges rovnyak spaces h b are star shaped we also prove a transfer principle which represents h b spaces inside d

the fredholm index of a pair of commuting operators ii this paper concerns fredholm theory in several variables and its applications to hilbert spaces of analytic functions one feature is the introduction of ideas from commutative algebra to operator theory specifically we introduce a method to calculate the fredholm index of a pair of commuting operators to achieve this we define and study the hilbert space analogs of samuel multiplicities in commutative algebra then the theory is applied to the symmetric fock space in particular our results imply a satisfactory answer to arveson s program on developing a fredholm theory for pure d contractions when d 2 including both the fredholmness problem and the calculation of indices we also show that arveson s curvature invariant is in fact always equal to the samuel multiplicity for an arbitrary pure d contraction with finite defect rank it follows that the curvature is a similarity invariant

existence of non subnormal polynomially hyponormal operators in 1950 p r halmos motivated in part by the successful development of the theory of normal operators introduced the notions of subnormality and hyponormality for bounded hilbert space operators an operator t is subnormal if it is the restriction of a normal operator to an invariant subspace t is hyponormal if t t tt it is a simple matrix calculation to verify that subnormality implies hyponormality but the converse is false one reason is that subnormality is invariant under polynomial calculus indeed analytic functional calculus while hyponormality is not if one then defines t to be polynomially hyponormal when p t is hyponormal for every polynomial p e c z the following question arises naturally

a gleason kahane żelazko theorem for the dirichlet space abstract we show that every linear functional on the dirichlet space that is non zero on nowhere vanishing functions is necessarily a multiple of a point evaluation continuity of the functional is not assumed as an application we obtain a characterization of weighted composition operators on the dirichlet space as being exactly those linear maps that send nowhere vanishing functions to nowhere vanishing functions we also investigate possible extensions to weighted dirichlet spaces with superharmonic weights as part of our investigation we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane

on convex cyclic operators abstract we give a hahn banach characterization for convex cyclicity we also obtain an example of a bounded linear operator s on a banach space with p s such that s is convex cyclic but s is not weakly hypercyclic and s 2 is not convex cyclic this solved two questions of rezaei in 25 when p s we also characterize the diagonalizable normal operators that are convex cyclic and give a condition on the eigenvalues of an arbitrary operator for it to be convex cyclic we show that certain adjoint multiplication operators are convex cyclic and show that some are convex cyclic but no convex polynomial of the operator is hypercyclic also some adjoint multiplication operators are convex cyclic but not 1 weakly hypercyclic

cyclic vectors in the dirichlet space we study the hilbert space of analytic functions with finite dinchlet integral in the open unit disc we try to identify the functions whose polynomial multiples are dense in this space theorems 1 and 2 confirm a special case of the following conjecture if ij z i ig z l at all points and if g is cyclic thenjis cyclic theorems 3 5 give a sufficient condition t is an outer function with some smoothness and the boundary zero set is at most countable and a necessary condition the radial limit can vanish only for a set of loganthmic capacity zero for a function j to be cyclic introduction in this paper we shall study the hilbert space of analytic functions in the open unit disc a in the complex plane that have a finite dirichlet integral jj if t12 dx dy g z l for some cyclic g and all z question 4 asks if f must be cyclic whenever f and l f are both in the space no examples are known where either of these questions has a negative answer in 2 we begin the study of cyclic vectors in the dirichlet space d theorems 1 and 2 give a partial answer to question 3 above for this space this section also contains 2 propositions 10 11 and 4 questions 7 10 proposition 11 says that if f and g are bounded functions in d whose product is cyclic then bothf and g must be cyclic theorem 2 gives a partial converse we require that gl be dini continuous on received by the editors july 21 1983 1980 mathematics subject classification primary 30h05 secondary 46e15 46e20 47b37

ergodic and dynamical properties of m isometries abstract an example of a weakly ergodic 3 isometry is provided in 3 we give new examples of weakly ergodic 3 isometries and study numerically hypercyclic m isometries on finite and infinite dimensional hilbert spaces in particular all weakly ergodic strict 3 isometries on a hilbert space are weakly numerically hypercyclic adjoints of unilateral forward weighted shifts which are strict m isometries on l 2 n are shown to be hypercyclic

extremal functions as divisors for kernels of toeplitz operators in this paper an extremal function of a banach space of analytic functions in the unit disk not all functions vanishing at 0 is a function solving the extremal problem supref 0 for functions f of norm 1 we study extremal functions of kernels of toeplitz operators on hardy spaces hp 1 2 modulo p dependent multiplicative constants we give examples showing that the extremal function may fail to be a contractive divisor when p 2 and also fail to be an expansive divisor when p 2 we discuss to what extent these results characterize the toeplitz operators via invariant subspaces for the backward shift

factorizations induced by complete nevanlinna pick factors abstract we prove a factorization theorem for reproducing kernel hilbert spaces whose kernel has a normalized complete nevanlinna pick factor this result relates the functions in the original space to pointwise multipliers determined by the nevanlinna pick kernel and has a number of interesting applications for example for a large class of spaces including dirichlet and drury arveson spaces we construct for every function f in the space a pluriharmonic majorant of f 2 with the property that whenever the majorant is bounded the corresponding function f is a pointwise multiplier

properties of hermite and laguerre polynomials in matrix argument and their applications abstract this paper is concerned with the hermite polynomials in symmetric and rectangular matrix argument and the laguerre polynomials in symmetric matrix argument which are associated with the symmetric and rectangular matrix variate normal distributions and the wishart distribution respectively we derive the multivariate rodrigues formula a differential equation for each of the two kinds of hermite polynomials with respect to the associating p d f and the mutivariate differential operator i e the zonal polynomial in the matrix of differential operators which plays the role of the ordinary differential the multivariate rodrigues formulae are useful in asymptotic multivariate distribution theory for matrix variates we consider the general forms of the series edgeworth expansions for distributions of both symmetric and rectangular random matrices and then give an asymptotic edgeworth expansion for the distribution of a scale mixture matrix as a simple example as for the laguerre polynomials alternative forms of differential equations are derived we also present series expansions and recurrence relationships for the hermite and laguerre polynomials and some limit properties of the laguerre polynomials

regularity for generators of invariant subspaces of the dirichlet shift abstract let d denote the classical dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable for a set e d we write d e f d lim r 1 f r e i t 0 q e where q e stands for except possibly for e i t in a set of logarithmic capacity 0 we show that if e is a carleson set then there is a function f d e that is also in the disc algebra and that generates d e in the sense that d e clos p f p is a polynomial we also show that if d is an extremal function i e p p 0 for every polynomial p then the limits of z exist for every e i t d as z approaches e i t from within any polynomially tangential approach region

on projections in a space with an indefinite metric abstract we consider g projectors orthogonal projections defined on an indefinite inner product space and derive in a systematic way the indefinite counterparts of a number of useful results known to hold for ordinary projectors in hilbert space some of the topological considerations encountered in the literature are avoided here and several results are obtained using quite elementary matrix type arguments in particular the relation between g projectors and contractions in an indefinite inner product space is studied for example a convergence result is given for a nondecreasing sequence of g contractive g projectors we also prove a result characterizing g projectors within the class of idempotents generalizing the corresponding result in hilbert space

weighted shift and composition operators on lp which are m q isometries abstract we obtain some conditions about the weighted shift and the composition operators on l p to be m q isometries in particular we prove that if t is a strict 2 q isometry then p q whether t is a weighted shift or a composition operator on l p moreover we characterize the admissible weights for the weighted shift to be a strict 3 p isometry on l p the case of strict 4 p isometry is also studied

on the effects of moderate multivariate nonnormality on roy s largest root test abstract the asymptotic distribution of roy s largest root statistic in multivariate edgeworth populations is expanded to terms of the first order to this order the effects of non normality are expressed by mardia s measures of multivariate skewness and kurtosis together with a supplementary skewness measure tables of corrections to the nominal significance level are presented for the k sample manova situation comparison with simulation results indicates that the correction terms provide a useful indication both of the magnitude and the direction of the effect of nonnormality on the nominal 5 percent type i error even when the underlying population is not well represented by an edgeworth expansion provided that the skewness and kurtosis are not too large

generalized hermite and laguerre polynomials in multiple symmetric matrix arguments and their applications abstract we investigate properties of the generalized hermite h m r x q b r and laguerre l u q r x q b r polynomials in q m m symmetric matrix arguments x 1 x q x q and r m m constant matrices b 1 b r b r and with q parameters u 1 u q u q q r and consider their applications in multivariate distribution theory the results extend those for the case of one symmetric matrix argument obtained by chikuse we derive the generalized multivariate rodrigues formula a differential equation for the polynomial h m r with respect to the associated joint multiple symmetric matrix variate normal p d f and the generalized multivariate differential operator i e the invariant polynomial in the multiple matrices of differential operators which plays the role of the ordinary differential the generalized multivariate rodrigues formula and the associated polynomials h m r are utilized to present the general form of the series edgeworth expansion for a distribution of multiple symmetric random matrices and then give a simple example alternative forms of differential equations are derived for the polynomial l u q r series expansions and recurrence relationships or some related results are obtained for the polynomials h m r and l u q r

products of orthogonal projections and polar decompositions abstract we characterize the sets x of all products pq and y of all products pqp where p q run over all orthogonal projections and we solve the problems arg min p q p q z for z x or y we also determine the polar decompositions and moore penrose pseudoinverses of elements of x

isometric properties of elementary operators abstract we consider the elementary operator l acting on the hilbert schmidt class c 2 h given by l t atb with a and b bounded operators on h we establish necessary and sufficient conditions on a and b for l to be a 2 isometry or a 3 isometry we derive sufficient conditions for l to be an n isometry we also give several illustrative examples involving the weighted shift operator on l 2 and the multiplication operator on the dirichlet space

analysis of linear combinations with extreme ratios of variance abstract if the covariance matrices 1 and 2 of two multivariate populations are not identical insight into the differences between 1 and 2 can often be gained by analyzing the linear combinations with extreme ratios of variances that is those defined by the eigenvectors associated with the extreme roots of 1 1 2 this article gives a descriptive method similar to variable selection procedures in regression for screening and simplifying these linear combinations a hypothesis of redundancy of variables is defined and a statistic for testing this hypothesis is derived the method is illustrated by an example

wold type decompositions and wandering subspaces for operators close to isometries for operators t satisfying certain inequalities we obtain a weak analog of the classical wold kolmogorov decomposition theorem representing t as a direct sum of a unitary operator and a shift operator acting in some hilbert space of analytic functions the concept of a dual operator is introduced which reflects relationships between shift operators acting in two hilbert spaces of analytic functions dual to each other with respect to the cauchy pairing among different aspects of this duality we consider relationships between hereditary inequalities and properties of reproducing kernels

unbounded or bounded idempotent operators in hilbert space abstract a densely defined closed linear operator f in a hilbert space is said to be idempotent if ran f dom f and f f f we show that such an idempotent operator is written as f p p q 1 2 p q 1 2 where p and q are the orthoprojections to ran f and ker f respectively when f is bounded this becomes f p p q 1 further we show that for any 0 the operator p q is invertible and f p p q 1 in addition to the known results we present several descriptions of the norm f in terms of p q p q 1 or p q 1

multivariate meixner classes of invariant distributions abstract we define multivariate meixner classes of invariant distributions of random matrices as those whose generating functions for the associated orthogonal polynomials are of certain special integral or summation forms generalizing the univariate meixner classes of distributions which were first characterized by meixner 21 characterization theorems and properties of these multivariate meixner classes are established the zonal polynomials the extended invariant polynomials with matrix arguments and their related results in multivariate distribution theory are utilized in the discussion

on walsh differentiable dyadically stationary random processes some basic properties of dyadically stationary ds processes are introduced including continuity and spectral representation a sampling theorem based on the walsh functions is investigated for random signals that are not necessarily sequency limited by using the concept of a dyadic derivative the resulting aliasing error is calculated together with the speed of convergence an example gives a glimpse into the possibilities of applying the sampling theorem as well as the dyadic derivative

c0 semigroups of m isometries on hilbert spaces abstract let t t t 0 be a c 0 semigroup on a separable hilbert space h we show that t t is an m isometry for any t if and only if the mapping t r t t x 2 for each x h is a polynomial of degree at most m this property is used to study m isometric right translation semigroup on weighted l p spaces we also provide alternative characterizations of the above property by imposing conditions on the infinitesimal generator operator and on the cogenerator operator of t t t 0 moreover we prove that a non unitary 2 isometry t on a hilbert space satisfying the kernel condition that is t t k e r t k e r t can be embedded into a c 0 semigroup if and only if d i m k e r t