lipschitz operator ideals and the approximation property abstract we establish the basics of the theory of lipschitz operator ideals with the aim of recovering several classes of lipschitz maps related to absolute summability that have been introduced in the literature in the last years as an application we extend the notion and main results on the approximation property for banach spaces to the case of metric spaces

phase transitions in simple learning the authors investigate learning in the simplest type of a layered neural network the one layer perceptron the learning process is treated as a statistical dynamical problem quantities one is interested in include the relaxation time the learning time and the capacity and how they depend on noise and constraints on the weights the relaxation time is calculated as a function of the noise level and the number p of associations to be learned they consider three different cases for input patterns that are random and uncorrelated in the first where the connection weights are constrained to satisfy n 1 sigma i omega i2 s2 there is a critical value of p n separating regimes of perfect and imperfect learning at zero noise in contrast the second model unconstrained learning exhibits a different kind of transition at p n and noise plays no role in the third model where the constraint is imposed only on the thermal fluctuations there is a line of phase transitions terminating at p n and zero noise they have also considered learning with correlated input patterns the most important difference is the emergence of a second relaxation time which the authors interpret as the time it takes to learn a prototype of the patterns

structure of metastable states in the hopfield model an upper bound for the number of metastable states in the hopfield model is calculated as a function of the hamming fraction from an input pattern for all finite values of alpha the ratio of number of patterns to nodes the hamming fraction from the input pattern to the nearest metastable state is infinite when alpha 0 113 the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the hamming fraction 0 5 from it

probing the basins of attraction of a recurrent neural network a recurrent neural network is considered that can retrieve a collection of patterns as well as slightly perturbed versions of this pure set of patterns via fixed points of its dynamics by replacing the set of dynamical constraints i e the fixed point equations by an extended collection of fixed point like equations analytical expressions are found for the weights wij b of the net which depend on a certain parameter b this so called basin parameter b is such that for b 0 there are a priori no perturbed patterns to be recognized by the net it is shown by a numerical study via probing sets that a net constructed to recognize perturbed patterns i e with values of the connections wij b with b 0 possesses larger basins of attraction than a net made with the help of a pure set of patterns i e with connections wij b 0 the mathematical results obtained can in principle be realized by an actual biological neural net

operator p compact mappings abstract we introduce the classes of operator p compact mappings and completely right p nuclear operators which are natural extensions to the operator space framework of their corresponding banach operator ideals we relate these two classes define natural operator space structures and study several properties of these ideals we show that the class of operator compact mappings in fact coincides with a notion already introduced by webster in the nineties in a very different language this allows us to provide an operator space structure to webster s class

learning and retrieval in attractor neural networks above saturation the authors investigate neural networks in the range of parameters when the ground state energy is positive namely when a synaptic matrix which satisfies all the desired constraints cannot be found by the learning algorithm in particular they calculate the typical distribution functions of local stabilities obtained for a number of algorithms in this region these functions are used to investigate the retrieval properties as reflected by the size of the basins of attraction this is done analytically in sparsely connected networks and numerically in fully connected networks the main conclusion is that the retrieval behaviour of attractor neural networks can be improved by learning above saturation

the nature of attractors in an asymmetric spin glass with deterministic dynamics the authors study the attractors in an infinite range ising spin glass model with deterministic dynamics where the interactions have asymmetry specified by a parameter k they find a duality relation between the attractors for models with asymmetry parameters k and 1 k the attractors are fixed points or limit cycles of short length except for k 1 at which the average cycle length diverges reminiscent of a phase transition and the model has many similarities to the random map model as well as to the infinite range symmetric spin glass in thermal equilibrium including the fact that a few attractors dominate the weight the extent of this dominance varies from sample to sample and so is given by a non trivial probability distribution pi y which they compute numerically

k sample anderson darling tests abstract two k sample versions of an anderson darling rank statistic are proposed for testing the homogeneity of samples their asymptotic null distributions are derived for the continuous as well as the discrete case in the continuous case the asymptotic distributions coincide with the k 1 fold convolution of the asymptotic distribution for the anderson darling one sample statistic the quality of this large sample approximation is investigated for small samples through monte carlo simulation this is done for both versions of the statistic under various degrees of data rounding and sample size imbalances tables for carrying out these tests are provided and their usage in combining independent one or k sample anderson darling tests is pointed out the test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample one weighting adjusts for the possibly different s

content addressability and learning in neural networks the content addressability of patterns stored in ising spin neural network models with symmetric interactions is studied numerical results from simulations on the icl distributed array processor dap involving systems with up to 2048 neurons are presented behaviour consistent with finite size scaling characteristic of a first order phase transition is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it

a remark on the approximation of p compact operators by finite rank operators abstract the banach operator ideal k p of p compact operators was introduced in p d sinha a k karn compact operators whose adjoints factor through subspaces of l p studia math 150 2002 17 33 let p 1 and let x be a closed subspace of an l p space we show that x has the approximation property if and only if for every banach space y the linear space f y x of finite rank operators is k p norm dense in k p y x i e x has the k p approximation property

metastable states of neural networks incorporating the physiological dale hypothesis physiologically it is likely that excitatory and inhibitory neurons are rather clearly distinguished or in other words each neuron has in most cases a unique excitatory or inhibitory property the dale hypothesis to study the consequence of the physiological constraint the author proposes a learning rule for neural networks which incorporates the constraint then the distribution of the metastable states is calculated and it is found that the retrieval states form a much larger group in the proposed model than in the hopfield model the author also studies the process of retrieval by considering the statistical dynamics of the overlaps the result suggests that excitatory neurons and inhibitory neurons are preferably balanced in number if the attraction basins of the stored patterns are to be sufficiently large

retrieval phase diagrams for attractor neural networks with optimal interactions the authors consider the retrieval properties of attractor neural networks whose synaptic matrices have been constructed to maximise the number of patterns which can be stored in a perceptron satisfying certain constraints retrieval is studied in the absence as well as in the presence of fast noise temperature the discussion is restricted to dilute networks for which dynamical equations for the overlaps are available when the patterns are stored with a prescribed lower limit on the stability parameter on every site the full stability of the perceptron ensures the existence of an attractor with perfect retrieval it is found that the curve of critical storage capacity alpha against temperature t as a line of first order transitions for high values of alpha and becomes second order for low alpha at a point of a tricritical nature the phase diagram is compared with the dilute hopfield model it is found that at high synaptic noise levels the diluted hopfield net stores more effectively than the network trained for optimal perceptron storage when a given fraction of sites is allowed to violate the stability bound the solution of the perceptron learning problem does not ensure the existence of an attractor of finite overlap even in the absence of noise this case is studied separately for t 0 and for finite t

the roles of stability and symmetry in the dynamics of neural networks the authors study the retrieval phase of spin glass like neural networks considering that the dynamics should depend only on gauge invariant quantities they propose that two such parameters characterising the symmetry of the neural net s connections and the stabilities of the patterns are responsible for most of the dynamical effects this is supported by a numerical study of the shape of the basins of attraction for a one pattern neural network model the effects of stability and symmetry on the short time dynamics of this model are studied analytically and the full dynamics for vanishing symmetry is shown to be exactly solvable

the dual space of l x y τp and the p approximation property we establish a representation of the dual space of l x y the space of bounded linear operators from a banach space x into a banach space y endowed with the topology p of uniform convergence on p compact subsets of x we apply this representation and solve the duality problem for the p approximation property p ap that is if the dual space x has the p ap then so does x however the converse does not hold in general we show that given 2 2p p 2 this subspace is the davie space in lq davie 1973 5 which does not have the approximation property it follows that for every 2 p there exists a banach space yp such that it has the p ap but its dual space yp fails to have the p ap we study the relation of the p ap with the denseness of finite rank operators in the topology p finally we introduce the p compact approximation property p cap and show for every 2 p that the davie space in c0 fails to have the p cap and also that a variant of the willis space willis 1992 17 has the p cap but it fails to have the p ap

parallel dynamics of the neural network with the pseudoinverse coupling matrix the authors investigate the parallel dynamics of the neural network with the pseudoinverse coupling matrix based on an exact dynamic theory they develop an approximate treatment for long timescales the temporal development of the overlap of correlation functions and of the remanent magnetization are investigated they present results for deterministic and stochastic dynamics large scale numerical simulations supplement their analytical findings

categorization in the pseudo inverse neural network we investigate analytically the emergence of the categorization ability in the pseudo inverse attractor neural network more pointedly we consider the problem of learning an extensive number of concepts by storing a finite number of examples s of each concept we find that there is a critical value beyond which the categorization error as measured by the average fraction of unstable sites in the concepts decreases monotonically with s

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

nonparametric comparison of two histograms scholz and stephens 1987 journal of the american statistical association 82 918 924 proposed a nonparametric k sample anderson darling statistic for grouped data this note demonstrates that a partition of chi 2 method may give a more powerful nonparametric test particularly when alternatives other than location shift are important a taste test example provides motivation

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

learning with noise in a linear perceptron the learning of a set of p random patterns in a linear perceptron is studied in the limit of a large n of input units with noise on the weights inputs and output the problem is formulated in continuous time as a langevin equation and the first task is to evaluate the response or green function for the system white noise on the output is shown to correspond to spatially correlated weight noise acting only in a subspace of the weight space it is shown that the input noise acts as a simple weight decay with a size proportional to the load parameter alpha p n with no weight decay the relaxation time diverges at alpha 1 with a weight decay it becomes shorter and finite for alpha 1 but at the cost of a larger asymptotic learning error that is found analytically it is shown that a small weight decay decreases the effect of noise on the weights or outputs

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

retrieval properties of diluted attractor neural networks we investigate the dependence of the retrieval properties of the pseudo inverse and optimal attractor neural networks on the fraction of stored patterns the temperature t and the margin parameter phase diagrams in the full space of parameters are presented in the regime of extreme dilution i e when the connectivity c satisfies the condition where n is the number of neurons furthermore we study analytically the neighbourhood of a stored pattern for both models by calculating the average fraction of unstable sites in a pattern that differs by d sites from a given stored pattern this analysis may shed light on the properties of the basins of attraction of the stored patterns

polynomials and holomorphic functions on a compact sets in banach spaces abstract in this paper we study the behavior of holomorphic mappings on a compact sets motivated by the recent work of aron caliskan garcia and maestre 2016 we give several conditions on the holomorphic mappings and on the banach operator ideal a under which a compact sets are preserved appealing to the notion of tensor stability for operator ideals we first address the question in the polynomial setting then we define a radius of a b compactification that permits us to tackle the analytic case our approach for instance allows us to show that the image of any p r compact set under any holomorphic function defined on any open set of a banach space is again p r compact

comparing k populations with linear rank statistics abstract there are many nonparametric tests available for the analysis of k independent samples for example the kruskal wallis test wilcoxon rank sum test when k 2 is widely used for detecting location differences and mood or ansari bradley or sigel tukey rank statistics are often suggested for detecting scale differences if more general alternatives are of interest then one can use omnibus tests such as the generalized kolmogorov smirnov and cramer von mises statistics see conover 1980 secs 6 3 and 6 4 or kiefer 1959 unfortunately significant results from these omnibus tests are hard to interpret instead i would like to propose some new tests based on a 4 k table of linear rank statistics that are simple to interpret because the four rows of the table are aimed at specific alternatives location scale skewness and kurtosis row summary statistics consist of weighted sums of squares of the individual entries the first two of these are the kruskal wallis statistic and mood s k sa

nonparametric analysis for doubly ordered two way contingency tables tsujitani 1992 biometrics 48 267 269 and kateri 1993 biometrics 49 950 951 considered analyses for two way contingency tables in this reader reaction a nonparametric analysis using mnidrank scores is given this analysis can be interpreted in terms of bivariate moments does not make model assumptions takes into account the ordinal nature of the data and allows monte carlo p values to be given

equilibrium properties of the linear perceptron we study the equilibrium properties of the real weights linear perceptron within the replica formalism framework focusing on the effects of the normalization of the weights on the learning and generalization capabilities of the network we also investigate the effects of static noise corrupting the training data and of dynamical noise acting on the weights during the training stage

the number of metastable states in the projection rule neural network we calculate the direct average n of the number of metastable states in the projection rule neural network by means of the saddle point method n is obtained as a function of the hamming distance g to a given pattern stability gamma and energy e the critical storage capacity of the model is calculated the examination of the energy dependence of n leads to lower bounds for the energies of the spin glass states

on α nuclear operators with applications to vector valued function spaces abstract let x be a banach space and let y be a closed subspace of a banach space z let be a tensor norm our main result is as follows assume that x or z has the approximation property if there is a bounded linear extension operator from y to z then any bounded linear operator t x y is nuclear whenever t is nuclear from x to z using this result we characterize the space c p x respectively uc p x of continuous functions from a compact hausdorff space into a banach space x whose range is p compact respectively unconditionally p compact

optimal basins of attraction in randomly sparse neural network models the size of the basin of attraction for randomly sparse neural networks with optimal interactions is calculated for all values of the storage ratio alpha p c 2 where p is the number of random uncorrelated patterns and c is the connectivity the basin of attraction is finite while for alpha 0 42 the basin of attraction is almost 100

on p compact mappings and the p approximation property abstract the notion of p compact sets arises naturally from grothendieck s characterization of compact sets as those contained in the convex hull of a norm null sequence the definition due to sinha and karn 2002 leads to the concepts of p approximation property and p compact operators which form an ideal with its ideal norm p this paper examines the interaction between the p approximation property and certain space of holomorphic functions the p compact analytic functions in order to understand these functions we define a p compact radius of convergence which allows us to give a characterization of the functions in the class we show that p compact holomorphic functions behave more like nuclear than compact maps we use the product of schwartz to characterize the p approximation property of a banach space in terms of p compact homogeneous polynomials and in terms of p compact holomorphic functions with range on the space finally we show that p compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the p approximation property our approach also allows us to solve several questions posed by aron maestre and rueda 2010

multivalley structure of attractor neural networks we numerically investigate the structure of the basins of attraction and the nature of the spurious attractors of the pseudo inverse and the optimal weights attractor neural networks we show that the number of attractors in the optimal weights model increases as the margin parameter increases and that the basins of attraction of the stored patterns are not significantly enlarged by moreover the number of attractors is smaller than in the pseudo inverse model

density of finite rank operators in the banach space of p compact operators a banach space x is said to have the kp approximation property kp ap if for every banach space y the space f y x of finite rank operators is dense in the space kp y x of p compact operators endowed with its natural ideal norm kp in this paper we study this notion that has been previously treated by sinha and karn 2002 in 15 as application the kp ap of dual banach spaces is characterized via density of finite rank operators in the space of quasi p nuclear operators for the p summing norm this allows to obtain a relation between the kp ap and saphar s approximation property as another application the kp ap is characterized in terms of a trace condition finally we relate the kp ap to the p p approximation property introduced in sinha and karn 2002 15 for subspaces of lp spaces

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

learning algorithms with optimal stability in neural networks to ensure large basins of attraction in spin glass like neural networks of two state elements xi imu or 1 the authors propose to study learning rules with optimal stability delta where delta is the largest number satisfying delta or sigma j jij xi jmu xi imu mu 1 p i 1 n where n is the number of neurons and p is the number of patterns they motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix jij in addition numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns

grothendieckʼs nuclear operator theorem revisited with an application to p null sequences abstract let x and y be banach spaces and let be a tensor norm the principal result is the following theorem if either x or y has the approximation property then each nuclear operator t x y such that t y x can be approximated in the nuclear norm by finite rank operators of type x y in the special case of grothendieck nuclear operators the theorem provides a strengthening for the classical theorem on the nuclearity of operators with a nuclear adjoint the hypotheses about the approximation property are essential the main application yields an affirmative answer to c pineiro j m delgado p convergent sequences and banach spaces in which p compact sets are q compact proc amer math soc 139 2011 957 967 for p 1 a sequence x n x is p null if and only if lim x n 0 and x n is relatively p compact in x

on the ability of the optimal perceptron to generalise a linearly separable boolean function is derived from a set of examples by a perceptron with optimal stability the probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method

generalization in a linear perceptron in the presence of noise the authors study the evolution of the generalization ability of a simple linear perceptron with n inputs which learns to imitate a teacher perceptron the system is trained on p alpha n example inputs drawn from some distribution and the generalization ability is measured by the average agreement with the teacher on test examples drawn from the same distribution the dynamics may be solved analytically and exhibits a phase transition from imperfect to perfect generalization at alpha 1 when there are no errors static noise in the training examples if the examples are produced by an erroneous teacher overfitting is observed i e the generalization error starts to increase after a finite time of training it is shown that a weight decay of the same size as the variance of the noise errors on the teacher improves on the generalization and suppresses the overfitting the generalization error as a function of time is calculated numerically for various values of the parameters finally dynamic noise in the training is considered white noise on the input corresponds on average to a weight decay and can thus improve generalization whereas white noise on the weights or the output degrades generalization generalization is particularly sensitive to noise on the weights for alpha 1 where it makes the error constantly increase with time but this effect is also shown to be damped by a weight decay weight noise and output noise acts similarly above the transition at alpha 1

the phase space of interactions in neural networks with definite symmetry we calculate the typical fraction of the phase space of interactions which solve the problem of storing a given set of p patterns represented as n spin configurations as a function of the storage ratio a p n of the stability parameter k and of the symmetry 7 of the interaction matrices the calculation is performed for strongly diluted networks where the connectivity of each spin c is of the order of in n for each value of k and 17 there is a maximal value of a above which the volume of solutions vanishes for each value of k and a there is a typical value of 7 at which this volume is maximal the analytical studies are supplemented by numerical simulations on fully connected and diluted networks using specific learning algorithms

the p approximation property in terms of density of finite rank operators abstract we characterize the p approximation property p ap introduced by sinha and karn d p sinha a k karn compact operators whose adjoints factor through subspaces of l p studia math 150 2002 17 33 in terms of density of finite rank operators in the spaces of p compact and of adjoints of p summable operators as application the p ap of dual banach spaces is characterized via density of finite rank operators in the space of quasi p nuclear operators this relates the p ap to saphar s approximation property ap p as another application the p ap is characterized via a trace condition allowing to define the trace functional on certain subspaces of the space of nuclear operators

a replica analysis of the minimum weight solution to the linear equations problem we investigate through statistical mechanics techniques the problem of finding n tuples j ji j2 jn with a fixed fraction k of non vanishing entries that solve a set of p alpha n linear equations in particular we determine the region in the plane alpha k where these solutions exist with a probability of 1 in the limit of large n we also obtain at insight into the microscopic structure of these solutions by calculating the distribution of the probability of an arbitrary non vanishing entry ji moreover we evaluate analytically the performances of several easy to implement heuristic algorithms and compare them with the optimal solution a task that is suited very well to the statistical mechanics approach

multiconnected neural network models a generalisation of the hopfield model which includes interactions between p 2 ising spins is considered the exact storage capacity behaves as np 1 2 p 1 ln n when the number of nodes n is large in the limit p to infinity the thermodynamics of the model can be solved exactly without using the replica method at zero temperature a solution which is completely correlated with the input pattern exists for alpha alpha c where alpha c to infinity as p to infinity and this solution has lower energy than the spin glass solution if alpha alpha 1 1 4 ln 2 where the number of patterns n 2 alpha p np 1 for finite values of p the correlation with the input pattern is not complete for p 3 and 4 approximate values of alpha c and alpha 1 are obtained and for p to infinity the replica symmetric approximation gives alpha c approximately p 4 ln p

disease clustering in time tango 1984 biometrics 40 15 26 proposed a clustering index for testing for clusters of disease in time a test based on this clustering index was shown to compare favourably with other statistical tests in this article we show that pearson s x2 and its components perform well in testing for clusters of disease in time the rth of these components identifies a departure from uniformity in moments up to the rth and so helps describe the alternative if any to uniformity

the injective and surjective hulls of the ideal of p q compact operators and their approximation properties abstract let 1 p q with 1 p 1 q 1 we establish the isometric representations of the injective hull k i n j p q and surjective hull k p q s u r of the ideal of p q compact operators we obtain some factorizations of k p q i n j and k p q s u r and prove that if a banach space x has the k p p i n j approximation property k p p i n j ap respectively k p p s u r ap then x has the k p q i n j ap respectively k p q s u r ap it also follows that if a banach space x has the ap then x has the k p q i n j ap and the k s u r p q ap and that every banach space has the k i n j 2 q ap the k p 2 i n j ap k s u r 2 q ap and the k p 2 s u r ap finally we find the dual space of k p q i n j x y k p q i n j respectively k p q s u r x y k p q s u r when x has the ap or y has the k p q i n j ap respectively k p q s u r ap

the banach ideal of a compact operators and related approximation properties abstract we use the notion of a compact sets determined by an operator ideal a introduced by carl and stephani 1984 to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework for banach operator ideals a we introduce a way to measure the size of a compact sets and use it to give a norm on k a the ideal of a compact operators then we study two types of approximation properties determined by a compact sets we focus our attention on an approximation property which makes use of the norm defined on k a this notion fits the definition of the a approximation property recently introduced by oja 2012 with k a instead of a we exemplify the power of the carl stephani theory and the geometric structure introduced here by appealing to some recent developments on p compactness