global integral criteria for composition operators abstract let d denote the dirichlet type space of functions analytic on the unit disk u and q the conformal invariant version of this space any analytic self map of u induces a composition operator c acting on d respectively q by c f f where f d respectively f q the aim of this paper is to characterize boundedness and compactness of such operators in terms of global area integrals of

maps completely preserving commutativity and maps completely preserving jordan zero product abstract let x y be real or complex banach spaces with infinite dimension and let a b be standard operator algebras on x and y respectively in this paper we show that every map completely preserving commutativity from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism every map completely preserving jordan zero product from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism

additive mappings on von neumann algebras preserving absolute values abstract given hilbert spaces h and k and a von neumann algebra a b h let denote the class of all additive mappings a b k satisfying a a a a the paper shows that if a contains no nonzero abelian central projection then every preserves the operation the r linear combination and up to a commuting operator multiple i 0 the ring multiplication if a contains a nonzero abelian central projection p and if the dimension of k is at least 2 or 2 rank p according to whether or not p can be chosen to be minimal then there exists an additive mapping a b k such that i is a projection and a a for all a a but is neither multiplicative nor adjoint preserving in case a b h the result was proved by molnar bull aust math soc 53 1996 391 when a contained all finite rank operators and by radjabalipour et al linear algebra appl 327 2001 197 under the redundant restriction i i k i k

on maps preserving lie products equal to a rank one nilpotent abstract let be a bijective linear map on the algebra of n n complex matrices such that e 12 e 12 and a b e 12 whenever a b e 12 the purpose of this paper is to describe surprisingly has a different description from maps preserving zero lie products

commutativity preserving linear maps and lie automorphisms of triangular matrix algebras abstract in this article we classify linear maps from the algebra t n of n upper triangular matrices into itself satisfying ab ba 0 if and only if ab ba 0 in particular we show that for n 2 any such map is either the sum of an algebra automorphism and a map into the centre or the sum of the negative of an algebra anti automorphism and a map into the centre as a consequence lie automorphisms of the algebra t n are also classified

mappings that preserve pairs of operators with zero triple jordan product abstract let f be a field and n 3 suppose s 1 s 2 m n f contain all rank one idempotents the structure of surjections s 1 s 2 satisfying aba 0 a b a 0 is determined similar results are also obtained for a subsets of bounded operators acting on a complex or real banach space x b the space of hermitian matrices acting on n dimensional vectors over a skew field d c subsets of self adjoint bounded linear operators acting on an infinite dimensional complex hilbert space it is then illustrated that the results can be applied to characterize mappings on matrices or operators such that f aba f a b a for all a b for functions f such as the spectral norm schatten p norm numerical radius and numerical range etc

penalty barrier multiplier methods for convex programming problems we study a class of methods for solving convex programs which are based on nonquadratic augmented lagrangians for which the penalty parameters are functions of the multipliers this gives rise to lagrangians which are nonlinear in the multipliers each augmented lagrangian is specified by a choice of a penalty function varphi and a penalty updating function pi the requirements on varphi are mild and allow for the inclusion of most of the previously suggested augmented lagrangians more importantly a new type of penalty barrier function having a logarithmic branch glued to a quadratic branch is introduced and used to construct an efficient algorithm convergence of the algorithms is proved for the case of pi being a sublinear function of the dual multipliers the algorithms are tested on large scale quadratically constrained problems arising in structural optimization

commuting maps on rank k matrices abstract let n 2 be a natural number let m n k be the ring of all n n matrices over a field k fix natural number k satisfying 1 k n under a mild technical assumption over k we will show that additive maps g m n k m n k such that g x x 0 for every rank k matrix x m n openk are of form x x where z m n k z and z stands for the center of m n k furthermore we shall see an example that there are additive maps such that g x x 0 for all rank 1 matrices that are not of the form x x

linear transformations on symmetric matrices that preserve commutativity it is shown that if a nonsingular linear transformation t on the space of n square real symmetric matrices preserves the commutativity where n 3 then t a qaqt q a in for all symmetric matricesa for some scalar orthogonal matrix q and linear functional q

jordan mappings of semiprime rings an additive mapping 8 of a ring r into a 2 torsion free ring r is called a jordan homomorphism if 6 ab z a 8 b 6 b a for all a be r a well known result of i n herstein 4 states that every jordan homomorphism onto a prime ring is either a homomorphism or an antihomomorphism suppose that r contains ideals u and v with null intersection let cp r u be a homomorphism and ii r v be an antihomomorphism a mapping 6 40 is a so called direct sum of mappings rp and 11 obviously b is a jordan homomorphism according to this construction we see that herstein s result 4 does not hold in semip me rings moreover w e baxter and w s martindale el showed by an example that a jordan homomorphism 0 of a ring r onto a semiprime ring r is not necessarily a direct sum of a homomorphism and an antihomomorphism but they proved that there always exists an essential ideal e of r such that the restriction of 8 to e is a direct sum of a homomorphism cp e r and an antihomomorphism y5 e r 1 theorem 2 71 in this paper theorem 2 3 we extend this result by showing that e can be choosen so that it is a sum of ideals u and v of r such that cp vanishes on v and tc vanishes on u even more for each x e r we have b ux e u e x for all u e u and b ux f x e o for all u e v this result also contains an affirmative answer to the question of baxter and martindale whether there is a way to choose the ideal e so that 8 e is an associative subring of r as we shall see in our case 6 e is the essential associative ideal of r an additive mapping 0 of a ring r into a ring r which satisfies b aba a 6 b 6 a for all a b e r will be called a jordan triple homomorphism an easy computation shows that every jordan homomorphism is also a

jordan isomorphisms of upper triangular matrix rings abstract let r be a 2 torsionfree ring with identity 1 and let t n r n 2 be the ring of all upper triangular n n matrices over r we describe additive jordan isomorphisms of t n r onto an arbitrary ring and generalize several results on this line

the essential norm of a composition operator mapping into the qs space abstract an asymptotic formula in terms of a global integral condition for the essential norm c e of the composition operator c f f mapping from the weighted bergman space a p 1 p 2 or the weighted dirichlet space d into the mobius invariant q s space is established moreover it is shown that if c is bounded from the classical dirichlet space to bmoa then c e 2 lim sup a 1 sup b d d a z 2 n b z d a z lim sup z 1 sup b d n b z where n b z is the nevanlinna counting function for b and b z b z 1 b z

commuting maps on some subsets of matrices that are not closed under addition abstract let m n k be the ring of all n n matrices over a field k we describe additive maps g m n k m n k such that g x x xg x for all invertible singular x m n k

multiplicative lie n derivations of generalized matrix algebras abstract the aim of the paper is to give a description of multiplicative lie n derivations for a certain class of generalized matrix algebras as a consequence multiplicative lie n derivations of full matrix algebras are determined this solves a conjecture due to benkovic and eremita in 2012

generalized functional identities with anti automorphisms and derivations on prime rings i abstract the concept of a generalized functional identity gfi with anti automorphisms and derivations is a generalization of the notion of a generalized polynomial identity gpi with anti automorphisms and derivations in the present paper we show that either a prime ring is gpi or such gfis have only the obvious solutions our results simultaneously generalize the results of martindale rowen lanski kharchenko and chuang on gpis and a number of earlier results on gfis

invertible linear maps on borel subalgebras preserving zero lie products abstract let g be a simple lie algebra of rank l over an algebraically closed field of characteristic zero b a borel subalgebra of g an invertible linear map on b is said preserving zero lie products in both directions if for x y b x y 0 if and only if x y 0 in this paper it is shown that an invertible linear map on b preserving zero lie products in both directions if and only if it is a composition of an inner automorphism a graph automorphism a scalar multiplication map and a diagonal automorphism which extends the main result in 8 from a linear solvable lie algebra to far more general cases

generalized lie derivations on skew elements of prime algebras with involution we characterize generalized lie derivations on skew elements of prime algebras a with involution provided that a does not satisfy polynomial identities of low degree the analogous result for matrix algebras is also described

lie derivations on banach algebras abstract let d be a lie derivation on a unital complex banach algebra a then for every primitive ideal p of a except for a finite set of them which have finite codimension greater than one there exist a derivation d from a p to itself and a linear functional on a such that q p d a d a p a for all a a where q p denotes the quotient map from a onto a p moreover the preceding decomposition holds for all primitive ideals in the case where d is continuous

commuting traces and commutativity preserving maps on triangular algebras abstract let a be a triangular algebra the problem of describing the form of a bilinear map b a a a satisfying b x x x x b x x for all x a is considered as an application commutativity preserving maps and lie isomorphisms of certain triangular algebras e g upper triangular matrix algebras and nest algebras are determined

strong commutativity preserving maps on lie ideals abstract let a be a prime ring and let r be a noncentral lie ideal of a an additive map f r a is called strong commutativity preserving scp on r if f x f y x y for all x y r in this paper we show that if f is scp on r then there exist c 2 1 and an additive map r z a such that f x x x for all x r where c is the extended centroid of a unless char a 2 and a satisfies the standard identity of degree 4

commuting traces of biadditive mappings commutativity preserving mappings and lie mappings biadditive mappings b r r r where r is a prime ring with certain additional properties satisfying b x x x xb x x for all x r are characterized as an application we determine the structures of commutativity preserving mappings lie isomorphisms and lie derivations of certain prime rings

generalized lie derivations on triangular algebras let a be a unital algebra and let m be a unitary a bimodule we consider generalized lie derivations mapping from a to m our results are applied to triangular algebras in particular to nest algebras and block upper triangular matrix algebras we prove that under certain conditions each generalized lie derivation of a triangular algebra a is the sum of a generalized derivation and a central map which vanishes on all commutators of a

lagrangian duality and related multiplier methods for variational inequality problems we consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities developing a simple lagrangian duality scheme which is combined with the recent logarithmic quadratic proximal lqp theory introduced by the authors we derive three algorithms for solving the variational inequality vi problem this provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms we prove primal dual and primal dual convergence under very mild assumptions eliminating all the usual assumptions used until now in the literature for related algorithms applications to complementarity problems are also discussed

on surjective linear maps preserving commutativity we describe surjective linear maps preserving commutativity from symmetric elements of any algebra with involution onto symmetric elements of a prime algebra with involution not satisfying polynomial identities of low degree bijective commutativity preservers on skew elements of centrally closed prime algebras with involution of the first kind are also investigated

a globally convergent newton method for solving strongly monotone variational inequalities variational inequality problems have been used to formulate and study equilibrium problems which arise in many fields including economics operations research and regional sciences for solving variational inequality problems various iterative methods such as projection methods and the nonlinear jacobi method have been developed these methods are convergent to a solution under certain conditions but their rates of convergence are typically linear in this paper we propose to modify the newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length the purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate it is then shown that under the strong monotonicity assumption the method is globally convergent and under some additional assumptions the rate of convergence is quadratic limited computational experience indicates the high efficiency of the proposed method

convergence of proximal like algorithms we analyze proximal methods based on entropy like distances for the minimization of convex functions subject to nonnegativity constraints we prove global convergence results for the methods with approximate minimization steps and an ergodic convergence result for the case of finding a zero of a maximal monotone operator we also consider linearly constrained convex problems and establish a quadratic convergence rate result for linear programs our analysis allows us to simplify and extend the available convergence results for these methods

linear transformations on self adjoint matrices the preservation of rank one plus scalar abstract let m be the space of self adjoint linear maps for a nondegenerate quadratic form on a finite dimensional vector space characteristic 2 dimension at least 3 let t m m be an invertible linear transformation if t preserves the set of rank one plus scalar maps then t is of the form t x rpxp x i with one exception over z 3 z any t preserving commuting pairs also has this form with no exceptions the proof of this latter result involves computing the dimension of the space of self adjoint maps commuting with any given one

on herstein s lie map conjectures i first published in transactions american mathematical society in vol 353 no 10 pp 4235 4260 published by the american mathematical society

estimates of essential norms of weighted composition operators between bloch type spaces abstract we estimate the essential norm of weighted composition operators u c acting on bloch type spaces b in terms of the analytic function u d c and the n th power of the analytic selfmap of the open unit disc d we obtain new characterizations for boundedness and compactness of u c b b for all 0 thus answering an open problem of manhas and zhao concerning the case 1

nonlinear strong commutativity preserving maps on skew elements of prime rings with involution abstract let a be a prime ring of characteristic not 2 with involution with center z a and with skew elements k suppose that f k a is a map satisfying f x f y x y for all x y k then there exists a map k z a such that f x x x for all x k or f x x x for all x k except when a is an order in a 4 9 or 16 dimensional central simple algebra

on maps preserving zeros of the polynomial xy yx abstract let a m n f be the matrix algebra over a field f with an involution where n 20 suppose that a a is a bijective linear map such that x y y x for all x y a such that xy yx we show that is of the form x uxu 1 for x a where is a nonzero symmetric scalar and u is a normal matrix such that uu is a nonzero scalar

on bilinear maps on matrices with applications to commutativity preservers abstract let m n be the algebra of all n n matrices over a commutative unital ring c and let l be a c module various characterizations of bilinear maps m n m n l with the property that x y 0 whenever x any y commute are given as the main application of this result we obtain the definitive solution of the problem of describing not necessarily bijective commutativity preserving linear maps from m n into m n for the case where c is an arbitrary field moreover this description is valid in every finite dimensional central simple algebra

strong commutativity preserving maps in prime rings with involution abstract let a be a prime ring of characteristic not 2 with center z a and with involution let s be the set of symmetric elements of a suppose that f s a is an additive map such that f x f y x y for all x y s then unless a is an order in a 4 dimensional central simple algebra there exists an additive map s z a such that f x x x for all x s or f x x x for all x s

lie isomorphisms in prime rings with involution abstract let r and r be prime rings with involutions of the first kind and with respective lie subrings of skew elements k and k furthermore assume rc c 1 4 9 16 25 64 where c is the extended centroid of r it is shown that any lie isomorphism of k onto k can be extended uniquely to an associative isomorphism of k onto k where k and k are respectively the associative subrings generated by k and k

strong commutativity preserving maps on subsets of matrices that are not closed under addition abstract let m n d be the ring of all n n matrices over a division ring d where n 2 is an integer and let gl n d be the set of all invertible matrices in m n d we describe maps f gl n d m n d such that f x f y x y for all x y gl n d the analogous result for singular matrices is also obtained

linear maps preserving commutativity abstract commutativity preserving maps on the real space of all real symmetric or complex self adjoint matrices are characterized related results are given for adjoint preserving maps defined on all n n matrices these results are extended to infinite dimensions in the case of invertible maps

maps on classes of hilbert space operators preserving measure of commutativity abstract in this paper first we give a partial answer to a question of l molnar and w timmermann namely we will describe those linear not necessarily bijective transformations on the set of self adjoint matrices which preserve a unitarily invariant norm of the commutator after that we will characterize those not necessarily linear or bijective maps on the set of self adjoint rank one projections acting on a two dimensional complex hilbert space which leave the latter quantity invariant finally this result will be applied in order to obtain a description of such bijective preservers on the unitary group and on the set of density operators

commutativity preserving linear maps on central simple algebras abstract let f be a field with char f 0 and let a be a finite dimensional central simple algebra over f with dim f a 4 if a linear map a a satisfies x 2 x x x 2 for all x a in particular if preserves commutativity then is either a standard commutativity preserving map or its range is commutative

lie derivable maps on b x abstract let x be a banach space of dimension greater than 1 we prove that if a map b x b x satisfies a b a b a b for any a b b x then d where d is an additive derivation of b x and the map b x f i vanishes at commutators a b

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

power commuting additive maps on invertible or singular matrices abstract let f be a field and let m n f be the ring of all n n matrices over f where n 2 is an integer we characterize additive maps f m n f m n f satisfying f x x m x m f x for all invertible singular x m n f where m 2 is an integer

herstein s lie theory revisited an associative ring r becomes a jordan ring r under x 0 y xy yx and a lie ring r under x y xy yx in case r has an involution i e an antiautomorphism of period 1 or 2 then the set of symmetric elements s s e r 1 s s is a jordan subring of r and the set of skew elements k ker 1 k k is a lie subring of rp in the early 1950s herstein initiated a study of the jordan and lie ideals of r s and k in case that r was a simple associative ring either without or with an involution in the ensuring years his work has been generalized in various directions on the one hand to the setting of prime and semiprime rings and on the other hand to invariance conditions other than that given by ideals besides herstein himself we mention lanski as having been a major force in this program their influence is certainly felt in the present paper we have taken the liberty of using various arguments without making specific mention from their papers notably 6 and s other important contributions in this area have also been made by baxter chacron erickson montgomery osborn streb and others part of our motivation in writing this paper is to obtain the lie ideal theory for semiprime rings with involution by a somewhat different approach from the self contained elementary very clever methods embodied in the original style of herstein let us describe very briefly the spirit of our approach for r a ring with involution and 1 any ideal i e an ideal invariant under in k k is always a lie ideal of k which we shall call a standard lie ideal of k a

centralizing maps on invertible or singular matrices over division rings abstract let d be a division ring and let m n d be the ring of all n n matrices over d with center z where n 2 is an integer we describe the additive map f m n d m n d such that f x x x f x z for all invertible singular x m n d

the structure of nonlinear lie derivation on von neumann algebras national natural science foundation of china 11071201 youth national natural science foundation of china 11001230 youth talented natural science foundation of fujian 2008f3103 fundamental research funds for the central universities 2010121001

additive mappings on operator algebras preserving absolute values abstract it is shown that an additive map b h b k is the sum of two homomorphisms one of which is c linear and the other is c antilinear provided that 1 a a a for all a b h 2 b i is an orthogonal projection and 3 c i i k i k the structure of is more refined when it is injective the paper also studies the properties of in the absence of condition b here b h and b k denote the algebras of all bounded linear operators on hilbert spaces h and k respectively these extend a result of l molnar bull austral math soc 53 1996 391 saying an additive map b h b h is a constant multiple of an either c linear or c antilinear homomorphism provided that 1 a a a for all a b h and 2 b b h contains all finite rank operators

commutativity preserving operators on symmetric matrices abstract if is a nonsingular linear operator on n n symmetric matrices over a formally real field n 3 and if preserves commutativity then a cu 1 au f a i where u t u and c are scalar and is a linear functional this is an extension of a result of chan and lim

new estimates of essential norms of weighted composition operators between bloch type spaces abstract we give estimates of the essential norms of the weighted composition operators u c between different bloch type spaces in terms of the n th power of we also give similar characterizations for boundedness and compactness of u c between different bloch type spaces

on additive isomorphisms of prime rings preserving polynomials we study additive isomorphisms of prime rings preserving a multilinear polynomial of degree 2 our main theorem generalizes a number of results obtained for jordan n jordan lie and lie triple isomorphisms of prime rings

essential norm estimates for composition operators on bmoa abstract we provide two function theoretic estimates for the essential norm of a composition operator c acting on the space bmoa one in terms of the n th power n of the symbol and one which involves the nevanlinna counting function we also show that if the symbol is univalent then the essential norm of c is comparable to its essential norm on the bloch space

lie triple derivations of triangular algebras abstract let r be a commutative ring with identity a b be unital algebras over r and m be a unital a b bimodule which is faithful as a left a module and also as a right b module let t a m 0 b be the triangular algebra consisting of a b and m this work is motivated by some intensive works of bresar 4 cheung 9 and zhang et al 30 here we study lie triple derivations of t it is shown that under mild assumptions every lie triple derivation on t is of standard form that is l can be expressed through an additive derivation and a linear functional vanishing on all second commutators of t examples of lie triple derivations on some classical triangular algebras are supplied

on operators preserving commutativity abstract let l x be the algebra of all bounded operators on a non trivial complex banach space x and f l x l x a bijective linear operator such that f and f 1 both send commuting pairs of operators into commuting pairs then either f a uau 1 p a i or f a ua u 1 p a i where p is a linear functional on l x u is a bounded linear bijective operator between the appropriate two spaces is a complex constant and a is the adjoint of a the form of an operator f for which f and f 1 both send projections of rank one into projections of rank one is also determined

centralizing maps in prime rings with involution abstract let r be a prime ring with involution of characteristic 2 with center z skew elements k and extended centroid c theorem suppose k k 0 and k k is an additive map such that x x z for all x k then unless r is an order in a 16 dimensional central simple algebra there exists c and an additive map k c such that x x x for all x k

zero product determined matrix algebras abstract let a be an algebra over a commutative unital ring c we say that a is zero product determined if for every c module x and every bilinear map a a x the following holds if x y 0 whenever xy 0 then there exists a linear operator t such that x y t xy for all x y a if we replace in this definition the ordinary product by the lie resp jordan product then we say that a is zero lie resp jordan product determined we show that the matrix algebra m n b n 2 where b is any unital algebra is always zero product determined and under some technical restrictions it is also zero jordan product determined the bulk of the paper is devoted to the problem whether m n b is zero lie product determined we show that this does not hold true for all unital algebras b however if b is zero lie product determined then so is m n b

an self adaptive lqp method for constrained variational inequalities abstract in this paper we present a logarithmic quadratic proximal lqp type prediction correction methods for solving constrained variational inequalities vi s f where s is a convex set with linear constraints the computational load in each iteration is quite tiny however the number of iterations is significantly dependent on a parameter which balances the primal and dual variables we then propose a self adaptive prediction correction method that adjusts the scalar parameter automatically under certain conditions the global convergence of the proposed method is established in order to demonstrate the efficiency of the proposed method we provide numerical results for a convex nonlinear programming and traffic equilibrium problems

commutativity preserving maps we briefly describe several techniques that have been developed to solve the problem of characterizing commutativity preserving maps we prove a new result describing the general form of injective continuous commutativity preserving maps on full matrix algebras

commutativity preserving maps on lie ideals of prime algebras abstract one of the most studied linear preserver problems is the problem of characterizing maps preserving commutativity in this paper we study maps preserving commutativity on a lie ideal l of a prime algebra a we show by an example that in general we cannot expect the same standard conclusion as in the case when l a therefore we confine ourselves to some special classes of algebras where the usual result can be proved

commutativity preserving linear maps and lie automorphisms of strictly triangular matrix space abstract in this paper we classify linear maps preserving commutativity in both directions on the space n f of strictly upper triangular n 1 n 1 matrices over a field f we show that for n 3 a linear map on n f preserves commutativity in both directions if and only if f where is a product of standard maps on n f and f is a linear map of n f into its center

on maps preserving zeros of lie polynomials of small degrees we describe maps preserving zeros of multilinear lie polynomials of degrees 3 and 4 on prime algebras and matrices over unital algebras in particular our theorems generalize several results related to commutativity preserving maps

commuting traces of multiadditive mappings abstract letrbe a prime ring andmann additive mapping onrsuch that m x x x 0 for allx r if char r 0 or n then there exist mappings iofrinto the extended centroidcofrsuch that m x x i 0 n i x x n i for allx r if in addition ris not algebraic of bounded degree n then for eachithere exists ani additive mappingmiofriintocsuch that i x mi x x for allx r

on functional identities in prime rings with involution abstract let a be a prime ring with involution let s be the symmetric elements let k be the skew elements let q ml be the maximal left ring of quotients x 1 x m noncommuting variables and e i f j g k h l a m 1 q ml i j k l 1 2 m we study functional identities of the form m i 1 e i i x i m j 1 x j f j j m k 1 g k k x k m l 1 x l h l l 0 for all x 1 x m a where e i i means e i x 1 x i x m etc in case s k is not algebraic of bounded degree 2 m definitive results are obtained as an application k commuting traces of symmetric n additive maps of either s or k into q ml are characterized

idempotent elements determined matrix algebras abstract let m n r be the algebra of all n n matrices over a unital commutative ring r with 2 invertible v be an r module it is shown in this article that if a symmetric bilinear map from m n r m n r to v satisfies the condition that u u e u whenever u 2 u then there exists a linear map f from m n r to v such that x y f x y x y m n r applying the main result we prove that an invertible linear transformation on m n r preserves idempotent matrices if and only if it is a jordan automorphism and a linear transformation on m n r is a jordan derivation if and only if it is jordan derivable at all idempotent points

the essential norm of a weighted composition operator on bmoa we consider weighted composition operators w psi varphi f mapsto psi f circ varphi on the space bmoa which consists of the analytic functions on the unit disc whose boundary values have bounded mean oscillation on the unit circle we provide an estimate for the essential norm of w psi varphi in terms of the weight function psi and the n th power varphi n of the analytic self map varphi of the open unit disc we also provide a new estimate for the norm of w psi varphi