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Abstract We prove a Jensen-disc type theorem for polynomials p ∈ R [ z ] having all their zeros in a sector of the complex plane. This result is then used to prove the existence of a collection of linear operators T : R [ z ] → R [ z ] which map polynomials with their zeros in a closed convex sector | arg ⁡ z | ≤ θ < π / 2 to polynomials with zeros in a smaller sector | arg ⁡ z | ≤ γ < θ. We, therefore, provide the first example of a zero-sector reducing operator. [ABSTRACT FROM AUTHOR]

A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in R n , the unit ball of some norm is equal to the unit ball of the dual norm, then the norm must be Euclidean. Motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators, we consider, in a real Hilbert space setting, a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator. We show that if the linear operator is positive definite, then the considered equation is uniquely solvable by an ellipsoid. Otherwise, the equation can have several (possibly infinitely many) solutions or no solution at all. Our analysis yields a few by-products of possible independent interest, among them results related to coercive bilinear forms (essentially a quantitative convex analytic converse to the celebrated Lax–Milgram theorem from partial differential equations) and a characterization of real Hilbertian spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Kirchhoff type equations (0.1) { − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u + λ V ( x ) u = q ( x ) f ( u ) in R 3 , u ∈ H 1 ( R 3 ) , where a , b , λ > 0 , V ∈ C ( R 3 , R ) is a potential well, q ( x ) is a positive bounded function, f ( s ) is either asymptotically linear or asymptotically 3-linear in s at infinity. Under some other suitable conditions on V , q and f , the existence, nonexistence and concentrating behavior of solutions to problem (0.1) are obtained by using variational methods. We mainly extend the results in J. Sun and T. Wu (2014) [26] , which dealt with Kirchhoff type equations with positive potential well, to Kirchhoff type equations with sign-changing potential well. [ABSTRACT FROM AUTHOR]

We first review realizations of Herglotz functions in the unit ball of C N and provide new insights. Then, we define the corresponding class and prove the extend the results in the case of several quaternionic variables. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain–Brezis–Mironescu type limit formula for a related class of anisotropic nonlocal norms. [ABSTRACT FROM AUTHOR]

Dugundji spaces were introduced by Pełczyński as compact Hausdorff spaces X such that every embedding of X into a Tychonoff cube [ 0 , 1 ] A admits a linear extension operator u : C ( X ) → C ( [ 0 , 1 ] A ) such that ‖ u ‖ = 1 and u ( 1 X ) = 1 [ 0 , 1 ] A , where 1 X is the constant function on X taking value 1. In this paper we show that a compact space X is Dugundji provided that there exists a linear extension operator u : C ( X ) → C ( [ 0 , 1 ] A ) satisfying one of the following conditions: (a) ‖ u ‖ < 2 and | u ( f ⋅ g ) | ≤ ‖ g ‖ ⋅ | u ( | f | ) | for all f , g ∈ C ( X ) ; (b) ‖ u ‖ = 1 . [ABSTRACT FROM AUTHOR]

We show local solvability in Besov spaces for a class of first order linear operators L defined on an open set of R n + 1 , n ∈ N , satisfying the condition ( P ) of Nirenberg–Treves and whose coefficients are Hölder continuous. Moreover, when n = 1 , we show local solvability for L in L ∞ ( R , B ∞ , ∞ s ( R ) ) , B ∞ , ∞ s ( R 2 ) and L q ( R , B p , q s ( R ) ) , 1 < p < ∞ , 1 ≤ q ≤ ∞ , s ∈ R . Recalling that C s = B ∞ , ∞ s , if s > 0 and not an integer (Hölder space), then we have local solvability for L in L ∞ ( R , C s ( R ) ) and C s ( R 2 ) . [ABSTRACT FROM AUTHOR]

Our main target in this paper is to present new sharp bounds for inequalities that result when weighted operator means are filtered through positive linear maps and operator monotone functions. As an application, we prove a refined reverse of the celebrated Golden–Thompson inequality. Furthermore, we show how these inequalities can be squared. [ABSTRACT FROM AUTHOR]

When studying non-symmetric nonlocal operators on R d : L f ( x ) = ∫ R d ( f ( x + z ) − f ( x ) − ∇ f ( x ) ⋅ z 1 { | z | ⩽ 1 } ) κ ( x , z ) | z | d + α d z , where 0 < α < 2 , d ⩾ 1 , and κ ( x , z ) is a function on R d × R d that is bounded between two positive constants, it is customary to assume that κ ( x , z ) is symmetric in z . In this paper, we study heat kernel of L and derive its two-sided sharp bounds without the symmetric assumption κ ( x , z ) = κ ( x , − z ) . In fact, we allow the kernel κ to be time-dependent and x → κ ( t , x , z ) to be only locally β -Hölder continuous with Hölder constant possibly growing at a polynomial rate in | z | . We also derive gradient estimate when β ∈ ( 0 ∨ ( 1 − α ) , 1 ) as well as fractional derivative estimate of order θ ∈ ( 0 , ( α + β ) ∧ 2 ) for the heat kernel. Moreover, when α ∈ ( 1 , 2 ) , drift perturbation of the time-dependent non-local operator L t with drift in Kato's class is also studied in this paper. As an application, when κ ( x , z ) = κ ( z ) does not depend on x , we show the boundedness of nonlocal Riesz's transformation: for any p > 2 d / ( d + α ) , ‖ L 1 / 2 f ‖ p ≍ ‖ Γ ( f ) 1 / 2 ‖ p , where Γ ( f ) : = 1 2 L ( f 2 ) − f L f is the carré du champ operator associated with L , and L 1 / 2 is the square root operator of L defined by using Bochner's subordination. Here ≍ means that both sides are comparable up to a constant multiple. [ABSTRACT FROM AUTHOR]

A bounded linear operator T : H 1 → H 2 , where H 1 , H 2 are Hilbert spaces is said to be norm attaining if there exists a unit vector x ∈ H 1 such that ‖ T x ‖ = ‖ T ‖ . If for any closed subspace M of H 1 , the restriction T | M : M → H 2 of T to M is norm attaining, then T is called an absolutely norm attaining operator or AN -operator. We prove the following characterization theorem: a positive operator T defined on an infinite dimensional Hilbert space H is an AN -operator if and only if the essential spectrum of T is a single point and [ m ( T ) , m e ( T ) ) contains atmost finitely many points. Here m ( T ) and m e ( T ) are the minimum modulus and essential minimum modulus of T , respectively. As a consequence we obtain a sufficient condition under which the AN -property of an operator implies AN -property of its adjoint. We also study the structure of paranormal AN -operators and give a necessary and sufficient condition under which a paranormal AN -operator is normal. [ABSTRACT FROM AUTHOR]

We find lower bounds for the set of Lipschitz constants of a given Lipschitzian map, defined on the closed unit ball of a Hilbert space, with respect to any renorming. We introduce a class of maps, defined in the closed unit ball of ℓ 2 , which contains the classical fixed point free maps due to Goebel–Kirk–Thelle, Baillon, and P.K. Lin. We show that for any map of this class its uniform Lipschitz constant with respect to any renorming of ℓ 2 is never strictly less than π 2 . [ABSTRACT FROM AUTHOR]

The paper aims at constructing two different solutions to an elliptic system u ⋅ ∇ u + ( − Δ ) m u = λ F defined on the two dimensional torus. It can be viewed as an elliptic regularization of the stationary Burgers 2D system. A motivation to consider the above system comes from an examination of unusual properties of the linear operator λ sin ⁡ y ∂ x w + ( − Δ ) m w arising from a linearization of the equation about the dominant part of F . We argue that the skew-symmetric part of the operator provides in some sense a smallness of norms of the linear operator inverse. Our analytical proof is valid for a particular force F and for λ > λ 0 , m > m 0 sufficiently large. The main steps of the proof concern finite dimension approximation of the system and concentrate on analysis of features of large matrices, which resembles standard numerical analysis. Our analytical results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

Let X = ( X , B , μ ) be a σ -finite measure space and f : X → X be a measurable transformation such that the composition operator T f : φ ↦ φ ∘ f is a bounded linear operator acting on L p ( X , B , μ ) , 1 ≤ p < ∞ . We provide a necessary and sufficient condition on f for T f to be topologically transitive or topologically mixing. We also characterize the topological dynamics of composition operators induced by weighted shifts, non-singular odometers and inner functions. The results provided in this article hold for composition operators acting on more general Banach spaces of functions. [ABSTRACT FROM AUTHOR]

Let W α be a bounded weighted shift with weight sequence α = { α n } n = 0 ∞ and let γ n = α 0 2 ⋯ α n − 1 2 ( n ≥ 1 ) with γ 0 = 1 . The positivity of both of the infinite matrices ( γ i + j ) 0 ≤ i , j < ∞ and ( γ i + j + 1 ) 0 ≤ i , j < ∞ is a condition equivalent to subnormality of W α . For a positive integer n , the positivity of ( γ i + j ) 0 ≤ i , j < n defines property H ( n ) for W α which is closely related to the flatness of α . As a study of the flatness property, the problem “describe weighted shifts W α with property H ( n ) such that α 1 = α 2 ” is considered. We solve this problem in the case of weighted shifts whose weight sequence has Bergman tail, and also in the case of shifts whose weight sequence is the backward extension of a Hamburger completion ( α 0 , α 1 , α 2 ) H . In addition, we discuss some examples to show the properties H ( n ) , H ˜ ( n ) , and n -hyponormality are distinct. [ABSTRACT FROM AUTHOR]

Classes of homogeneous polynomials between Banach spaces have been studied in the last three decades from the perspective of the stability with respect to the composition of a homogeneous polynomial with linear operators. In an attempt to explore the underlying nonlinearity in a more consistent way and to bring the subject closer to its roots in Complex Analysis – and, also, taking into account recent results in the field – in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials. [ABSTRACT FROM AUTHOR]

We consider positive invertible Lamperti operators T f ( x ) = h ( x ) Φ f ( x ) such that Φ has no periodic part. Let A n , T be the sequence of averages of T and M T the ergodic maximal operator. It is obvious that if M T is bounded on some L p , 1 < p < ∞ , then sup ⁡ ‖ A n , T ‖ L p ( ν ) ≤ ‖ M T ‖ L p ( ν ) < ∞ . It is known that the converse is true. In this paper we search the sharp dependence of the norm ‖ M T ‖ L p ( ν ) with respect to sup n ⁡ ‖ A n , T ‖ L p ( ν ) < ∞ . We prove that ‖ M T ‖ L p ( ν ) ≤ C ( p ) ( sup n ∈ N ⁡ ‖ A n , T ‖ L p ( d ν ) ) p ′ , where p ′ = p / ( p − 1 ) is the conjugate exponent and C ( p ) depends only on p . Furthermore, the exponent p ′ is sharp. Our results are closely related to Buckley's theorem about sharp bounds for the Hardy–Littlewood maximal function. [ABSTRACT FROM AUTHOR]

Let G be a locally compact abelian group, let ν be a regular probability measure on G , let X be a Banach space, let π : G → B ( X ) be a bounded strongly continuous representation. Consider the average (or subordinated) operator S ( π , ν ) = ∫ G π ( t ) d ν ( t ) : X → X . We show that if X is a UMD Banach lattice and ν has bounded angular ratio, then S ( π , ν ) is a Ritt operator with a bounded H ∞ functional calculus. Next we show that if ν is the square of a symmetric probability measure and X is K -convex, then S ( π , ν ) is a Ritt operator. We further show that this assertion is false on any non K -convex space X . [ABSTRACT FROM AUTHOR]

The inverse spectral problem for a Sturm–Liouville problem which consists of a Sturm–Liouville equation defined on the interval [ 0 , a ] and two separated boundary conditions at the endpoints 0 and a , one of which is involved function f ( λ ) , is considered in this paper. When f ( λ ) is the type of Herglotz functions and is known as a priori, we prove that the potential on [ 0 , a ] and the other boundary condition can be uniquely determined in terms of appropriate partial information on the spectrum and partial information on the set of normalizing constants. [ABSTRACT FROM AUTHOR]

It is known that the Naimark complementary frames for a given frame are not necessarily unique up to the similarity. In this paper we introduce the concept of joint complementary frame pairs for a given dual frame pair, and prove that they are unique up to the joint similarity. As an application, we give a necessary and sufficient condition under which two Naimark complementary frames are similar. For different pairs of dual frames, we present an operator parameterization for their joint complementary frame pairs. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the existence and uniqueness of μ -pseudo almost periodic solutions to a class of semilinear parabolic evolution equations with inhomogeneous boundary conditions, under the assumption that the forcing terms of the equation are μ -pseudo almost periodic in the sense of Stepanov. We also provide a new composition result of μ -pseudo almost periodic functions in the sense of Stepanov. An example is given for illustration. [ABSTRACT FROM AUTHOR]

A standard way to obtain extensions (resp. liftings) of operators is by making the so-called operations of push-out (resp. pull-back). In this paper we study the preservation of some classes of operators associated with an operator ideal A under push-out extensions or pull-back liftings. We show several examples of classical operator ideals whose associated classes are preserved, we prove that the preservation of those classes under push-out extension or pull-back lifting implies that the space ideal of A satisfies the 3-space property, and we derive some results for A that are useful in the study of commutative diagrams of operators. [ABSTRACT FROM AUTHOR]

In this note we study the interpolation properties of complex symmetric operators on Hilbert spaces. We show that under certain conditions the complex symmetricity property is preserved under quadratic interpolation. We apply this result to the study of complex symmetric Toeplitz operators on weighted Hilbert spaces of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper we investigate the behavior of powers of operators with spectrum contained in the Cantor sets. As consequence we show that the Cantor sets satisfy spectral synthesis in a large class of weighted Beurling algebras. [ABSTRACT FROM AUTHOR]

We study the notion of a function almost periodic at infinity and present some spectral conditions of the almost periodicity at infinity for bounded solutions of a differential inclusions governed by closed multivalued linear operators in Banach spaces. Applications to degenerate differential equations and semilinear differential inclusions are given. As example, we consider the problem of stabilization of solutions to a heat equation. [ABSTRACT FROM AUTHOR]

We introduce the notions of strongly Lipschitz ( p , r , s ) -nuclear operators and strongly Lipschitz ( p , r , s ) -integral operators. We develop a theory of strongly Lipschitz ( p , r , s ) -nuclear operators and strongly Lipschitz ( p , r , s ) -integral operators which closely parallels the theory for linear operators. Their close relationships with other Lipschitz operator ideals are studied. [ABSTRACT FROM AUTHOR]

It is known (see the classical book “Linear Operators I” by N. Dunford and J.T. Schwarz [6] ) that compact nice operators from a Banach space X into the space of continuous functions C ( Z , R ) are extreme points of the unit ball of compact operators B ( K ( X , C ( Z , R ) ) ) , where Z is a compact Hausdorff space. Recall, that an operator T is called nice if T ⁎ ( Z ) ⊂ Ext B ( X ) , where the continuous mapping T ⁎ is defined by T ⁎ ( z ) ( x ) = T ( x ) ( z ) for all x ∈ X and z ∈ Z . It is evident that if the set Ext B ( X ) is closed, then T is nice if and only if T ⁎ ( Z 0 ) ⊂ Ext B ( X ) on a dense subset Z 0 ⊂ Z . In general the set of extreme points Ext B ( X ) need not to be closed. In 1992 A. Suarez-Granero and M. Wisła presented the criteria for closedness of the set of extreme points of the unit ball in the case of Orlicz spaces equipped with the Luxemburg norm. The aim of this paper is to present the criteria for the closedness of the set of extreme points of the unit ball for the wide class of norms – the so-called p -Amemiya norms. This class includes both the Orlicz and the Luxemburg norm. The Orlicz functions that generates Orlicz spaces are assumed to be as much general as possible, in particular the Orlicz functions can vanish outside zero and jump to infinity what follows that the investigated Orlicz spaces can contain isomorphic copy of L ∞ or be a subspace of L ∞ . [ABSTRACT FROM AUTHOR]

Let u be a nonconstant inner function. In this paper, we study the dual truncated Toeplitz operators on the orthogonal complement of the model space K u 2 . This is a new class of Toeplitz operator. We show the product of two dual truncated Toeplitz operators D f D g to be zero if and only if either f or g is zero. We give a necessary and sufficient condition for the product of two dual truncated Toeplitz operators to be a finite rank operator. Furthermore, a necessary and sufficient condition is found for the product of two dual truncated Toeplitz operators to be a dual truncated Toeplitz operator. The last two results are different from the classical Toeplitz operator theory. [ABSTRACT FROM AUTHOR]

In this paper we give a generalization of the problem that was posed by I. Raşa in [10] and was proved based on a probabilistic approach by Mrowiec, Rajba and Wąsowicz in [8] . We end this paper by proposing two open problems related to Raşa's proposed inequality. [ABSTRACT FROM AUTHOR]

In this paper, we provide several characterizations on weak recurrence of translation operators on weighted Lebesgue spaces and on continuous function spaces. It is shown that the translation operator admits a nonzero nonwandering point if and only if it has a nonzero recurrent point, if and only if it is hypercyclic; that the translation operator admits a nonzero almost periodic point if and only if it is Devaney chaotic. Moreover, every hypercyclic translation operator has infinite topological entropy, but the converse is not true. Finally, we remark that there exists a linear operator which has the specification property on the whole space. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the Bishop–Phelps–Bollobás property for numerical radius (BPBp- ν ) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of L ( L 1 ( μ ) ) have the BPBp- ν for every finite measure μ . As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on L 1 ( μ ) have the BPBp- ν . [ABSTRACT FROM AUTHOR]

A Banach space X is subprojective if every infinite-dimensional subspace of X has a subspace which is complemented in X . We prove that separable Nakano sequence spaces ℓ ( p n ) are subprojective. Subprojectivity is also characterized in separable Nakano function spaces L p ( ⋅ ) ( 0 , 1 ) and L p ( ⋅ ) ( 0 , ∞ ) . [ABSTRACT FROM AUTHOR]

In this paper we explore the properties of a bounded linear operator defined on a Banach space, in light of operator norm attainment. Using Birkhoff–James orthogonality techniques, we give a necessary condition for a nonzero bounded linear operator to attain norm at a particular point of the unit sphere. We prove four corollaries to establish the importance of our study. As part of our exploration, we also obtain a characterization of smooth Banach spaces in terms of operator norm attainment and Birkhoff–James orthogonality. Restricting our attention to l p 2 ( p ∈ N ∖ { 1 } ) spaces, we obtain an upper bound for the number of points at which any linear operator, which is not a scalar multiple of an isometry, may attain norm. [ABSTRACT FROM AUTHOR]

Let f ∈ L 1 [ 0 , 1 ] be a mean zero function and let f n , n = 1 , 2 , … , be the dyadic dilations and translations of f . We investigate conditions on f , under which the linear operator T f defined by T f h n = f n , n = 1 , 2 , … , where h n , n = 1 , 2 , … , are mean zero Haar functions, can be continuously extended to the closed linear span [ h n ] in a certain function space X . Among other results we prove that T f is bounded in every symmetric space with nontrivial Boyd indices whenever f ∈ B M O d and f has “good” Haar spectral properties. In the special case of so-called Haar chaoses the above results can be essentially refined and sharpened. In particular, we find necessary and sufficient conditions, under which the operator T f , generated by a Haar chaos f of order 1, is continuously invertible in L p for all 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

For Banach lattices E 1 , … , E m and F with 1-unconditional bases, we show that the monomial sequence forms a 1-unconditional basis of L r ( E 1 , … , E m ; F ) , the Banach lattice of all regular m -linear operators from E 1 × ⋯ × E m to F , if and only if each basis of E 1 , … , E m is shrinking and every positive m -linear operator from E 1 × ⋯ × E m to F is weakly sequentially continuous. As a consequence, we obtain necessary and sufficient conditions for which the m -fold Fremlin projective tensor product E 1 ⊗ ˆ | π | ⋯ ⊗ ˆ | π | E m (resp. the m -fold positive injective tensor product E 1 ⊗ ˇ | ϵ | ⋯ ⊗ ˇ | ϵ | E m ) has a shrinking basis or a boundedly complete basis. [ABSTRACT FROM AUTHOR]

Let A and B be positive contractions on a Hilbert space. It was known before that the numerical range W ( A B ) of their product is contained in the rectangular region [ − 1 / 8 , 1 ] × [ − 1 / 4 , 1 / 4 ] . In this paper, we determine exactly when W ( A B ) touches on its four sides. This follows from the more general information on a containing region of W ( A B ) and on when they share a common tangent line. We also give an example of A and B on three-dimensional space for which W ( A B ) is not symmetric with respect to the x -axis. [ABSTRACT FROM AUTHOR]

If g ∈ L 2 ( [ 0 , 2 π ] ) let g ˆ be the sequence of Fourier coefficients of g , let D denote differentiation and let I denote the identity operator. Given α , β ∈ Z , we consider the operator D 2 − i ( α + β ) D − α β I on the second order Sobolev space of L 2 ( [ 0 , 2 π ] ) . The multiplier of this operator is − ( n − α ) ( n − β ) considered as a function of n ∈ Z , so that g ˆ ( α ) = g ˆ ( β ) = 0 for any function g in the range of the operator. Let δ x denote the Dirac measure at x , and let ⁎ denote convolution. If b ∈ [ 0 , 2 π ] let λ b be the measure 2 − 1 [ ( e i b ( α − β 2 ) + e − i b ( α − β 2 ) ] δ 0 − 2 − 1 [ ( e i b ( α + β 2 ) δ b + e − i b ( α + β 2 ) δ − b ] . A function of the form λ b ⁎ f is called a generalised difference , and we let F be the family of functions h such that h is a sum of five generalised differences. It is shown that for g ∈ L 2 ( [ 0 , 2 π ] ) , g ∈ F if and only if g ˆ ( α ) = g ˆ ( β ) = 0 . Consequently, F is a Hilbert subspace of L 2 ( [ 0 , 2 π ] ) and it is the range of D 2 − i ( α + β ) D − α β I . The methods use partitions of intervals and estimates of integrals in Euclidean space. There are applications to the automatic continuity of linear forms in abstract harmonic analysis. [ABSTRACT FROM AUTHOR]

In this note, we prove that many von Neumann algebras can be generated by two unitary operators u and v with u 2 = v 3 = 1 . As applications, three important classes of II 1 factors possessing this property are those with Property Γ, or with a Cartan masa, or non-prime ones. [ABSTRACT FROM AUTHOR]

We investigate spectral properties of Markov semigroups in von Neumann algebras and their dual semigroups in a fairly general setting which assumes only the abelianess of the semigroups and positivity of the maps in question. In particular, we analyse various properties of the spectral subspaces, and relations between the spectra of the Markov semigroup and its dual semigroup. In our analysis, we make extensive use of ergodic and quasi-ergodic projections which seems to be a new but quite fruitful approach. [ABSTRACT FROM AUTHOR]

Let X be an abstract L -space and let Y be any Banach space. Motivated by a classical result of T. J. Abatzoglou that describes smooth points of the space of operators on a Hilbert space, we give a characterization of very smooth points in the space of operators from X to Y . We also show that a similar result can be proved when Y is an abstract M -space such that the set of extreme points of the dual unit ball is a weak ⁎ -compact set. [ABSTRACT FROM AUTHOR]

We suggest a somewhat new approach to the issue of Hyers–Ulam stability. Namely, let A , B be (real or complex) linear spaces, L : A → B be a linear operator, N : = k e r L , and ρ A and ρ B be semigauges on A and B , respectively. We say that L is HU-stable with constant K ≥ 0 if for each x ∈ A such that ρ B ( L x ) ≤ 1 there exists z ∈ N with ρ A ( z − x ) ≤ K . With that definition we obtain quite general outcomes concerning approximate solutions to some differential equations. [ABSTRACT FROM AUTHOR]

If the set of points where a function is nonzero is of finite measure, and its Weyl transform is a finite rank operator, then the function is identically zero. [ABSTRACT FROM AUTHOR]

We demonstrate new abstract characterizations for unital and non-unital operator spaces. We characterize unital operator spaces in terms of the cone of accretive operators (operators whose real part is positive). We show that matrix norms and accretive cones are induced by gauges, although inducing gauges are not unique in general. Finally, we show that completely positive completely contractive linear maps on non-unital operator spaces extend to any containing operator system if and only if the operator space is induced by a unique gauge. [ABSTRACT FROM AUTHOR]

In this work we present a necessary and sufficient condition for a class of abstract underdetermined systems to be solvable. We develop J. F. Trèves' ideas, presenting the so called condition ( ψ ) and its connection with the study of the solvability in consideration. We also prove the existence of finite order regularity solutions. [ABSTRACT FROM AUTHOR]

Let B ( H ) be the algebra of all bounded linear operators acting on a Hilbert space H . The main purpose in this paper is to obtain a characterization of bijective maps Φ : B ( H ) → B ( K ) , K Hilbert space, satisfying the following condition Δ λ ( Φ ( A ) ∘ Φ ( B ) ) = Φ ( Δ λ ( A ∘ B ) ) for all A , B ∈ B ( H ) , where Δ λ ( T ) stands the λ -Aluthge transform of the operator T ∈ B ( H ) and A ∘ B = 1 2 ( A B + B A ) is the Jordan product of A and B . We prove that a bijective map Φ satisfies the above condition, if and only if there exists a unitary linear bounded operator U : H → K , such that Φ has the form Φ ( A ) = U A U ⁎ for all A ∈ B ( H ) . [ABSTRACT FROM AUTHOR]

For a finite dimensional complex normed space X , we say that it has the simultaneous zero inclusion property if an invertible linear operator S on X has zero in its spatial numerical range if and only if zero is in the spatial numerical range of the inverse S − 1 , as well. We show that beside Hilbert spaces there are some other normed spaces with this property. On the other hand, space ℓ 1 ( n ) does not have this property. Since not every normed space has the simultaneous zero inclusion property, we explore the class of invertible operators at which this property holds. In the end, we consider a property which is stronger than the simultaneous zero inclusion property and is related to the question when it is possible, for every invertible operator S , to control the distance of 0 to the spatial numerical range of S − 1 by the distance of 0 to the spatial numerical range of S . [ABSTRACT FROM AUTHOR]

Let a linear map Φ between two unital C ⁎ -algebras be positive and unital. Kadison showed that if f ( t ) = | t | and Φ ( f ( X ) ) = f ( Φ ( X ) ) for all selfadjoint operators X , then Φ ( X 2 ) = Φ ( X ) 2 for all selfadjoint operators X , that is, Φ is a C ⁎ -homomorphism. Choi proved this fact for an operator convex function f , and then conjectured that this fact would hold for a non-affine continuous function f . We shall prove a refinement of his conjecture. Petz has further proved that if f ( Φ ( A ) ) = Φ ( f ( A ) ) for a non-affine operator convex function f and a fixed A , then Φ ( A 2 ) = Φ ( A ) 2 . Arveson called such a function f a rigid function . We shall directly show power functions t r are rigid functions on ( 0 , ∞ ) if r ≠ 0 , r ≠ 1 . [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and let L ( H ) be the algebra of all bounded linear operators on H . Descriptions of the condition spectra of various special classes of operators are established. As application, characterization is obtained for maps on L ( H ) leaving invariant the condition spectrum or the condition spectral radius of the product of operators. Analogous results are obtained for Jordan triple product of operators. [ABSTRACT FROM AUTHOR]

We let B be a separable Banach space, and let { Z n } be a sequence of independent and identically distributed random elements in B . Then we prove that for a given strongly periodic sequence of bounded linear operators { ρ n } , the order one autoregressive system equations X n = ρ n X n − 1 + Z n , n in set on integers, possesses a unique almost sure strictly periodically correlated solution; under E [ log + ⁡ ‖ Z 0 ‖ ] < ∞ , which appears to be necessary as well. We proceed on to derive the limiting distribution of ∑ n = 1 N X n that appears to be a Gaussian distribution on B . We also provide interesting examples and observations. [ABSTRACT FROM AUTHOR]

In this paper we propose a new over-relaxed variant of the hybrid proximal extragradient (HPE) algorithm, for the monotone inclusion problem, which uses a projection-free extragradient step with explicit over relaxed stepsize. Its global convergence as well as ergodic and nonergodic complexity rates are established. Moreover, local linear convergence rates are derived under some mild regularity condition. One benefit of the new over relaxed variant of the HPE is that it covers a large class of popular operator splitting methods and their over relaxed versions, thus providing a comprehensive insight on these operators splitting methods. In particular, forward Douglas Rachford splitting method, forward Spingarn's Partial Inverse method, forward Spingarn's partial inverse forward method and Davis–Yin's three operator splitting method are all included as special cases of the over relaxed HPE algorithm. Another benefit is that the interval of stepsize relaxation is easily estimated for these operator splitting methods under the presented framework. Additionally, the over relaxed Korpelevich's method and over relaxed forward–backward–forward method are formulated directly with convergence guarantee based on the proposed framework. The third benefit is that the local linear convergence for a large class of operator splitting methods is established effortlessly under metric subregularity condition. Moreover, this linear convergence condition is shown weaker than some existing ones that almost all require the strong monotonicity of the composite operators. [ABSTRACT FROM AUTHOR]