rotfel d type inequalities for norms abstract in this note we consider some norm inequalities related to the rotfel d trace inequality tr f a b tr f a f b for concave functions f 0 0 and arbitrary n by n matrices for instance we show that for a large class of non negative concave functions f t and for all symmetric norms we have f a b 2 f a f b and we conjecture that this holds for all non negative concave functions

interpolating the rotfel d inequality for unitarily invariant norms abstract let a and b be square matrices of the same order uchiyama proved that for all unitarily invariant norms f a b f a f b which extended the rotfel d trace inequality tr f a b tr f a tr f b for any concave function f 0 0 in this note we interpolate uchiyama s inequality by proving that for all unitarily invariant norms f a b f 1 2 a b a b a b a b f a f b for instance letting f t t yields a b 1 2 a b a b a b a b a b which interpolates the triangle inequality for norms

comparison of peak and rms gains for discrete time systems abstract a convolution system can have a frequency response which is small for all frequencies yet still greatly amplify the peaks of signals passing though it for finite dimensional systems however we establish the simple bound h 1 2 n 1 h h h where h 1 is the peak gain of the system h h is the maximum frequency response of the system and n its dimension the same result for continuous time systems is due to gohberg and doyle and it is mentioned in 6 the bound implies that h optimal controllers which minimize the maximum of some disturbance to error transfer function cannot have very large peak gains from the disturbance to error

examples of extreme cases in i 1 and h optimization abstract we are guided by the simple bound h l1 2n 1 h h established in 1 for finite dimensional systems where h l1 is the peak gain of the system and h h is the maximum frequency response of the system and n is the mcmillan degree to establish bounds on closed loop systems resulting from l1 and h minimization furthermore we illustrate that the effectiveness of adding frequency constraints in addition to the interpolation constraints in the l 1 minimization problem depends on the bandwidth of the plant

inequalities for sums and products of operators abstract let ai i 1 4 be compact operators on a complex separable hilbert space we show that 2 s j a 1 a 2 a 3 a 4 1 2 s j a 1 a 2 a 3 a 4 for j 1 2 from this we derive norm inequalities and investigate conditions for equality

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

blum hanson type ergodic theorems in noncommutative symmetric spaces we prove a version of blum hanson ergodic theorem for l1 l l1 l contractions acting in noncommutative symmetric spaces with order continuous norm a similar result is established for arbitrary contractions on hardy subspaces of p convex schatten ideals of compact operators

a matrix subadditivity inequality for f a b and f a f b abstract in 1999 ando and zhan proved a subadditivity inequality for operator concave functions we extend it to all concave functions given positive semidefinite matrices a b and a non negative concave function f on 0 f a b f a f b for all symmetric norms in particular for all schatten p norms the case f t t is connected to some block matrix inequalities for instance the operator norm inequality a x x b max x b x for any partitioned hermitian matrix

the limit sets of some infinitely generated schottky groups let p be a packing of balls in euclidean space e n having the property that the radius of every ball of p lies in the interval 1 k k if g is a schottky group associated to p then the hausdorff dimension of the topological limit set of g is less than a uniform constant c k n n in particular this limit set has zero volume

submodular spectral functions of principal submatrices of a hermitian matrix extensions and applications we extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions we show that if f is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix a then the function i mapsto rm tr f a i is supermodular meaning that rm tr f a i rm tr f a j leq rm tr f a i cup j rm tr f a i cap j where a i denotes the i times i principal submatrix of a we discuss extensions to self adjoint operators on infinite dimensional hilbert space and to m matrices we discuss an application to cur approximation of nonnegative hermitian matrices

complete characterization of hadamard powers preserving loewner positivity monotonicity and convexity abstract entrywise powers of symmetric matrices preserving positivity monotonicity or convexity with respect to the loewner ordering arise in various applications and have received much attention recently in the literature following fitzgerald and horn 1977 8 it is well known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness similar phenomena have also recently been shown by hiai 2009 to occur for monotonicity and convexity in this paper we complete the characterization of all the entrywise powers below and above the critical exponents that are positive monotone or convex on the cone of positive semidefinite matrices we then extend the original problem by fully classifying the positive monotone or convex powers in a more general setting where additional rank constraints are imposed on the matrices we also classify the entrywise powers that are super sub additive with respect to the loewner ordering finally we extend all the previous characterizations to matrices with negative entries our analysis consequently allows us to answer a question raised by bhatia and elsner 2007 regarding the smallest dimension for which even extensions of the power functions do not preserve loewner positivity

more accurate weak majorization relations for the jensen and some related inequalities abstract motivated by results of aujla and silva 3 we give several more precise weak majorization and eigenvalue inequalities for some matrix versions of the famous jensen inequality with regard to a convexity our main results are then applied to log convex functions as an application we obtain refinements of some well known matrix inequalities

norm inequalities related to operator monotone functions let a b be positive semidefinite matrices and cdot any unitarily invariant norm on the space of matrices we show f a f b geq f a b for any non negative operator monotone function f t on 0 infty and g a g b leq g a b for non negative increasing function g t on 0 infty with g 0 0 and g infty infty whose inverse function is operator monotone

superadditivity and derivative of operator functions abstract we will show that if i j a i a j 0 for bounded operators a i 0 i 1 2 n then g i a i i g a i for every operator convex function g t on 0 with g 0 0 in particular i a i log i a i i a i log a i if each a i is invertible let a b 0 and a be invertible then we will observe that the frechet derivative d g s a b is increasing on 0 s for every operator convex function g t on 0 if and only if a b b a 0

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

extension of rotfel d theorem abstract let f t be a non negative concave function on the positive half line given an arbitrary partitioned positive semi definite matrix we show that f a x x b f a f b for all symmetric i e unitarily invariant norms this characterization of concave functions extends a famous trace inequality of rotfel d tr f a b tr f a tr f b and contains several classical matrix inequalities

mixed l sub 1 h sub spl infin control of mimo systems via convex optimization we present a methodology for designing mixed l sub 1 h sub spl infin controllers for mimo systems these controllers allow for minimizing the worst case peak output due to persistent disturbances while at the same time satisfying an h sub spl infin norm constraint upon a given closed loop transfer function therefore they are of particular interest for applications dealing with multiple performance specifications given in terms of the worst case peak values both in the time and frequency domains the main results of the paper show that 1 contrary to the h sub 2 h sub spl infin case the l sub 1 h sub spl infin problem admits a solution in l sub 1 and 2 rational suboptimal controllers can be obtained by solving a sequence of problems each one consisting of a finite dimensional convex optimization and a four block h sub spl infin problem moreover this sequence of controllers converges in the l sub 1 topology to an optimum

further results on the optimal rejection of persistent bounded disturbances the problem of designing controllers that optimally reject persistent disturbances is studied the focus is on the case where the plant to be controlled has either zeros or poles on the stability boundary i e the unit circle in the discrete time case and the extended j omega axis in the continuous time case for the discrete time case the problem of minimizing a cost functional of the form f rg sub 1 where the transform g of g has some unit circle zeros is studied a previously published dual problem formulation is extended and it is shown that an optimal controller need not exist the construction of a sequence of suboptimal controllers whose performance approaches the unattainable infimum of the cost function is studied it is shown that two results which hold in the case of h sub infinity optimization do not hold in the presented situation specifically the introduction of unit circle zeros can increase the value of the infimum even when every unit circle zero of g is also a zero of f and a sequence of controllers constructed in an obvious fashion fails to be an optimizing sequence similar results are obtained for the continuous time case

inequalities between f a b and f a f b abstract the conjecture posed by aujla and silva j s aujla f c silva weak majorization inequalities and convex functions linear algebra appl 369 2003 217 233 is proved it is shown that for any m tuple of positive semidefinite n n complex matrices a j and for any non negative convex function f on 0 with f 0 0 the inequality f a 1 f a 2 f a m f a 1 a 2 a m holds for any unitarily invariant norm it is also proved that f a 1 f a 2 f a m f a 1 a 2 a m where f is a non negative concave function on 0 and is normalized

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers