We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The …
We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for 0≤α<1 coincides with the classical definitions on polynomials (up to a constant). Further, if α=1, the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations.
In this article, we interpolate the well-known Young's inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. …
In this article, we interpolate the well-known Young's inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey.
The study of the continuity of the farthest point mapping for uniquely remotal sets has been used extensively in the literature to prove the singletoness of such sets. In this …
The study of the continuity of the farthest point mapping for uniquely remotal sets has been used extensively in the literature to prove the singletoness of such sets. In this article, we show that the farthest point mapping is not continuous even if the set is remotal, rather than being uniquely remotal. Consequently, we obtain some generalizations of results concerning the singletoness of remotal sets. In particular, it is proved that a compact set admitting a unique farthest point to its center is a singleton, generalizing the well known result of Klee.
If every point in a normed space X admits a unique farthest point in a given bounded subset E, then must E be a singleton?. This is known as the …
If every point in a normed space X admits a unique farthest point in a given bounded subset E, then must E be a singleton?. This is known as the farthest point problem. In an attempt to solve this problem, we give our contribution toward solving it, in the positive direction, by proving that every such subset E in the sequence space `1 is a singleton.
In this article, we investigate some operator-norm inequalities related to some conjectures posed by Hayajneh and Kittaneh that are related to questions of Bourin regarding a special type of inequalities …
In this article, we investigate some operator-norm inequalities related to some conjectures posed by Hayajneh and Kittaneh that are related to questions of Bourin regarding a special type of inequalities referred to as subadditivity inequalities. While some inequalities are meant to answer these conjectures, other inequalities present reverse-type inequalities for these conjectures. Then, we present some new trace inequalities related to Heinz means inequality and use these inequalities to prove some variants of the aforementioned conjectures.
In this article we interpolate Young’s inequality using a delicate treatment of dyadics. Although there are other simple methods to prove these results, we present this new approach hoping to …
In this article we interpolate Young’s inequality using a delicate treatment of dyadics. Although there are other simple methods to prove these results, we present this new approach hoping to reveal more of the hidden properties of such inequalities.
In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem.
In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem.
In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem.
In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem.
In this article we interpolate Young’s inequality using a delicate treatment of dyadics. Although there are other simple methods to prove these results, we present this new approach hoping to …
In this article we interpolate Young’s inequality using a delicate treatment of dyadics. Although there are other simple methods to prove these results, we present this new approach hoping to reveal more of the hidden properties of such inequalities.
If every point in a normed space X admits a unique farthest point in a given bounded subset E, then must E be a singleton?. This is known as the …
If every point in a normed space X admits a unique farthest point in a given bounded subset E, then must E be a singleton?. This is known as the farthest point problem. In an attempt to solve this problem, we give our contribution toward solving it, in the positive direction, by proving that every such subset E in the sequence space `1 is a singleton.
The study of the continuity of the farthest point mapping for uniquely remotal sets has been used extensively in the literature to prove the singletoness of such sets. In this …
The study of the continuity of the farthest point mapping for uniquely remotal sets has been used extensively in the literature to prove the singletoness of such sets. In this article, we show that the farthest point mapping is not continuous even if the set is remotal, rather than being uniquely remotal. Consequently, we obtain some generalizations of results concerning the singletoness of remotal sets. In particular, it is proved that a compact set admitting a unique farthest point to its center is a singleton, generalizing the well known result of Klee.
In this article, we interpolate the well-known Young's inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. …
In this article, we interpolate the well-known Young's inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey.
In this article, we investigate some operator-norm inequalities related to some conjectures posed by Hayajneh and Kittaneh that are related to questions of Bourin regarding a special type of inequalities …
In this article, we investigate some operator-norm inequalities related to some conjectures posed by Hayajneh and Kittaneh that are related to questions of Bourin regarding a special type of inequalities referred to as subadditivity inequalities. While some inequalities are meant to answer these conjectures, other inequalities present reverse-type inequalities for these conjectures. Then, we present some new trace inequalities related to Heinz means inequality and use these inequalities to prove some variants of the aforementioned conjectures.
We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The …
We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for 0≤α<1 coincides with the classical definitions on polynomials (up to a constant). Further, if α=1, the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations.
In this article, we interpolate the well-known Young's inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. …
In this article, we interpolate the well-known Young's inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey.
In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one …
In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one is also analytic.
In this list the ct's are systems with b(a)<~N.The e of the system {~j+l is smaller than the e of the system aj.The m's correspond to e's such that 1/2"J …
In this list the ct's are systems with b(a)<~N.The e of the system {~j+l is smaller than the e of the system aj.The m's correspond to e's such that 1/2"J is the e of aj.The superscripts c and nc correspond to cancelled and non-cancelled and * takes the values c and nc.B and G should suggest bad and good.In this list ( 2) and ( 3) are subcases of (1), ( 4) is a subcase of (3), ( 5) and ( 6) are subcases of ( 4), ( 7) and ( 8) are subcases of (5).After listing the cases we now turn to the statements.We assume ~n Ihh=l, IhlopN <1.gn
The following theorem is proved.Let T be a bounded linear operator on an infinite-dimensional Hubert space H over the complex numbers and let p(z) Ψ 0 be a polynomial with …
The following theorem is proved.Let T be a bounded linear operator on an infinite-dimensional Hubert space H over the complex numbers and let p(z) Ψ 0 be a polynomial with complex coefficients such that p(T) is completely continuous (compact).Then T leaves invariant at least one closed linear subspace of H other than H or {0}.For p(z) = z 2 this settles a problem raised by P. R. Halmos and K. T. Smith.The proof is within the framework of Nonstandard Analysis.That is to say, we associate with the Hubert space H (which, ruling out trivial cases, may be supposed separable) a larger space, *H, which has the same formal properties within a language L. L is a higher order language but *H still exists if we interpret the sentences of L in the sense of Henkin.The system of natural numbers which is associated with *H is a nonstandard model of arithmetic, i.e., it contains elements other than the standard natural numbers.The problem is solved by reducing it to the consideration of invariant subspaces in a subspace of *H the number of'whose dimensions is a nonstandard positive integer.
An overview of some recent developments on the Invariant Subspace Problem This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical …
An overview of some recent developments on the Invariant Subspace Problem This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read’s Banach space counter-example involving a finitely strictly singular operator.
This paper aims to discuss some inequalities for unitarily invariant norms.We obtain several inequalities for unitarily invariant norms.
This paper aims to discuss some inequalities for unitarily invariant norms.We obtain several inequalities for unitarily invariant norms.
We study the commutativity of two Toeplitz operators whose symbols are quasihomogeneous functions. We give a relationship between this commutativity and the roots (or powers) of the Toeplitz operators. We …
We study the commutativity of two Toeplitz operators whose symbols are quasihomogeneous functions. We give a relationship between this commutativity and the roots (or powers) of the Toeplitz operators. We use this to characterize Toeplitz operators with symbols in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity Baseline left-parenthesis double-struck upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">D</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^{\infty }(\mathbb {D})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which commute with Toeplitz operators whose symbols are of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript i p theta Baseline r Superscript m"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>p</mml:mi> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">e^{ip\theta }r^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
In this paper, which is a sequel to our earlier paper [1], we slightly modify the methods used in [1] to produce a continuous linear operator on l1 with no …
In this paper, which is a sequel to our earlier paper [1], we slightly modify the methods used in [1] to produce a continuous linear operator on l1 with no nontrivial closed invariant subspace.
Abstract : A proof is presented of the theorem that if B is a Banach space and if T is a completely continuous operator in B, there then exist proper …
Abstract : A proof is presented of the theorem that if B is a Banach space and if T is a completely continuous operator in B, there then exist proper invariant subspaces of T. The proof assumes strong convergence, complete continuity in the sense that any bounded set is transformed by T into a set with compact closure, and a strictly convex norm. The same theorem is proved for a Hilbert space; the theorem utilizes the concepts of weak and strong convergenece of elements and operators. The simplifying feature of the latter proof is that the metric propjections coincide with the usual orthogonal projections.
Let p, q > 0 satisfy 1/p + 1/q = 1. We prove that for any pair A, B of n × n complex matrices there is a unitary matrix …
Let p, q > 0 satisfy 1/p + 1/q = 1. We prove that for any pair A, B of n × n complex matrices there is a unitary matrix U, depending on A, B, such that$$U*\left| {AB*} \right|U \leqslant {\left| A \right|^p}/p + {\left| B \right|^q}/q.$$
We construct several examples of positive definite functions, and use the positive definite matrices arising from them to derive several inequalities for norms of operators. 1991 Mathematics Subject Classification 42A82, …
We construct several examples of positive definite functions, and use the positive definite matrices arising from them to derive several inequalities for norms of operators. 1991 Mathematics Subject Classification 42A82, 47A63, 15A45, 15A60.
Abstract We give an affirmative answer to one of the questions posed by Bourin regarding a special type of inequality referred to as subadditivity inequalities in the case of the …
Abstract We give an affirmative answer to one of the questions posed by Bourin regarding a special type of inequality referred to as subadditivity inequalities in the case of the Hilbert–Schmidt and the trace norms. We formulate the solution for arbitrary commuting positive operators, and we conjecture that it is true for all unitarily invariant norms and all commuting positive operators. New related trace inequalities are also presented.
The celebrated Heinz inequality asserts that 2|||A1/2XB1/2|||⩽|||AνXB1-ν+A1-νXBν|||⩽|||AX+XB||| for X∈B(H), A,B∈B(H)+, every unitarily invariant norm |||·||| and ν∈[0,1]. In this paper, we present several improvement of the Heinz inequality by using …
The celebrated Heinz inequality asserts that 2|||A1/2XB1/2|||⩽|||AνXB1-ν+A1-νXBν|||⩽|||AX+XB||| for X∈B(H), A,B∈B(H)+, every unitarily invariant norm |||·||| and ν∈[0,1]. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function F(ν)=|||AνXB1-ν+A1-νXBν|||, some integration techniques and various refinements of the Hermite–Hadamard inequality. In the setting of matrices we prove that Aα+β2XB1-α+β2+A1-α+β2XBα+β2⩽1|β-α|∫αβAνXB1-ν+A1-νXBνdν⩽12AαXB1-α+A1-αXBα+AβXB1-β+A1-βXBβ, for real numbers α, β.
In this article, we introduce the concept of harmonically log-convex functions, which seems to be strongly connected to unitarily invariant norms.Then, we prove Hermite-Hadamard inequalities for these functions.As an application, …
In this article, we introduce the concept of harmonically log-convex functions, which seems to be strongly connected to unitarily invariant norms.Then, we prove Hermite-Hadamard inequalities for these functions.As an application, we present many inequalities for the trace operator and unitarily invariant norms.
We make a progress towards describing the commutants of Toeplitz operators with harmonic symbols on the Bergman space over the unit disk. Our work greatly generalizes several partial results in …
We make a progress towards describing the commutants of Toeplitz operators with harmonic symbols on the Bergman space over the unit disk. Our work greatly generalizes several partial results in the field.
The main aim of this article is to present inequalities for the numerical radius of commutators of positive matrices. Some of these inequalities are analogues of known inequalities for unitarily …
The main aim of this article is to present inequalities for the numerical radius of commutators of positive matrices. Some of these inequalities are analogues of known inequalities for unitarily invariant norms. In particular, variants, but weaker forms, of the well-known Heinz inequality and its generalizations are extended to the context of numerical radius.
Given a complex Borel measure $\mu$ with compact support in the complex plane $\mathbb {C}$ the sesquilinear form defined on analytic polynomials $f$ and $g$ by $B_\mu (f,g) = \int …
Given a complex Borel measure $\mu$ with compact support in the complex plane $\mathbb {C}$ the sesquilinear form defined on analytic polynomials $f$ and $g$ by $B_\mu (f,g) = \int f\bar g d\mu$, determines an operator $T_\mu$ from the space of such polynomials $\mathcal {P}$ to the space of linear functionals on $\overline {\mathcal {P}}$. This operator is called the Toeplitz operator with symbol $\mu$. We show that $T_\mu$ has finite rank if and only if $\mu$ is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.
Abstract There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition …
Abstract There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts.
In [2], Abu Omar and Kittaneh defined a new generalization of the numerical radius.That is, given a norm N(•) on B(H) , the space of bounded linear operators over a …
In [2], Abu Omar and Kittaneh defined a new generalization of the numerical radius.That is, given a norm N(•) on B(H) , the space of bounded linear operators over a Hilbert space H , andThey proved several properties and introduced some inequalities.We continue with the study of this generalized numerical radius and we develop diverse inequalities involving w N .We also study particular cases when N(•) is the p -Schatten norm with p > 1 .
In this paper, we prove that each of the following functions is convex on R: f(t) = wN(AtXA1?t ? A1?tXAt), g(t) = wN(AtXA1?t), and h(t) = wN(AtXAt) where A > …
In this paper, we prove that each of the following functions is convex on R: f(t) = wN(AtXA1?t ? A1?tXAt), g(t) = wN(AtXA1?t), and h(t) = wN(AtXAt) where A > 0, X ? Mn and N(.) is a unitarily invariant norm onMn. Consequently, we answer positively the question concerning the convexity of the function t ? w(AtXAt) proposed by in (2018). We provide some generalizations and extensions of wN(.) by using Kwong functions. More precisely, we prove the following wN(f(A)X1(A) + g(A)Xf (A)) ? wN(AX+XA) ? 2wN(X)N(A), which is a kind of generalization of Heinz inequality for the generalized numerical radius norm. Finally, some inequalities for the Schatten p-generalized numerical radius for partitioned 2 ? 2 block matrices are established, which generalize the Hilbert-Schmidt numerical radius inequalities given by Aldalabih and Kittaneh in (2019).