scholarly journals Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

2021 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Function ◽  
Hilbert Spaces ◽  
Error Bounds ◽  
Real Function ◽  
Positive Measure ◽  
Integral Transform ◽  
Positive Function ◽  
Complex Hilbert Space ◽  
Second Derivative

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transformwhere the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, thenandwhere is the second derivative of as a real function.Applications for power function and logarithm are also provided.

scholarly journals Operator Subadditivity of the 𝒟-Logarithmic Integral Transform for Positive Operators in Hilbert Spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Hilbert Space ◽  
Positive Operator ◽  
Hilbert Spaces ◽  
Positive Measure ◽  
Integral Transform ◽  
Positive Function ◽  
Complex Hilbert Space ◽  
Monotone Functions ◽  
Logarithmic Integral ◽  
Operator Monotone Functions

Abstract For a continuous and positive function w (λ), λ> 0 and µ a positive measure on [0, ∞) we consider the following 𝒟-logarithmic integral transform 𝒟 ℒ o g ( w , μ ) ( T ) : = ∫ 0 ∞ w ( λ ) 1 n ( λ + T λ ) d μ ( λ ) , \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)d\mu \left( \lambda \right),} where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among others that, if A, B > 0 with BA + AB ≥ 0, then 𝒟 ℒ o g ( w , μ ) ( A ) + 𝒟 ℒ o g ( w , μ ) ( B ) ≥ 𝒟 ℒ o g ( w , μ ) ( A + B ) . \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right). In particular we have 1 6 π 2 + di log ( A + B ) ≥ di log ( A ) + di log ( B ) , {1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right), where the dilogarithmic function dilog : [0, ∞) → ℝ is defined by di log ( t ) : = ∫ 1 t 1 n s 1 - s d s ,         t ≥ 0. {\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0. Some examples for integral transform 𝒟Log (·, ·) related to the operator monotone functions are also provided.

scholarly journals ON THE POWER QUANTUM COMPUTATION OVER REAL HILBERT SPACES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350001 ◽  
Author(s):  
MATTHEW McKAGUE
Keyword(s):  
Hilbert Space ◽  
Quantum Computation ◽  
Hilbert Spaces ◽  
Quantum Circuit ◽  
Complex Hilbert Space ◽  
Complexity Classes ◽  
Complex Numbers ◽  
Real Numbers ◽  
Single Qubit ◽  
Quantum Complexity

We consider the power of various quantum complexity classes with the restriction that states and operators are defined over a real, rather than complex, Hilbert space. It is well known that a quantum circuit over the complex numbers can be transformed into a quantum circuit over the real numbers with the addition of a single qubit. This implies that BQP retains its power when restricted to using states and operations over the reals. We show that the same is true for QMA (k), QIP (k), QMIP and QSZK.

Some results on generalized finite operators and range kernel orthogonality in Hilbert spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 318-325
Author(s):  
Nadia Mesbah ◽  
Hadia Messaoudene ◽  
Asma Alharbi
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Generalized Derivation ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Similarity Orbit

Abstract Let ℋ {\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ ( ℋ ) {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ {\mathcal{ {\mathcal H} }} . In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) \left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥ A X − X B − I ∥ ≥ 1 , for all X ∈ ℬ ( ℋ ) . \parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

scholarly journals Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

2019 ◽  
Vol 31 (04) ◽  
pp. 1950013 ◽  
Author(s):  
Valter Moretti ◽  
Marco Oppio
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Hilbert Spaces ◽  
Von Neumann Algebras ◽  
Complex Structure ◽  
Complex Hilbert Space ◽  
Von Neumann ◽  
The Real ◽  
Quaternionic Hilbert Space ◽  
Adjoint Operators

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

scholarly journals Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry

2017 ◽  
Vol 29 (06) ◽  
pp. 1750021 ◽  
Author(s):  
Valter Moretti ◽  
Marco Oppio
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Real Hilbert Space ◽  
Mass Operator ◽  
Complex Structure ◽  
Complex Hilbert Space ◽  
Relativistic System ◽  
The Real ◽  
Algebra Of Observables ◽  
Poincaré Symmetry

As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda–Maeda and other authors), from the lattice theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another — in our opinion, deeper — viewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the real one, all selfadjoint operators represent observables in accordance with Solèr’s thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras.

scholarly journals Orbits tending to infinity under sequences of operators on Hilbert spaces

Filomat ◽  
2007 ◽  
Vol 21 (2) ◽  
pp. 161-171
Author(s):  
Sonja Mancevska ◽  
Marija Orovcanec
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Complex Hilbert Space ◽  
Infinite Dimensional ◽  
Set Of Points ◽  
Dense Set

In this paper are considered some sufficient conditions under which, for given sequence (Ti)i?1 of operators on an infinite-dimensional complex Hilbert space, there is a dense set of points whose orbits under each Ti tend strongly to infinity. .

scholarly journals Aluthge Transformation and *- Aluthge Transformation on M class Ak* Operator

2019 ◽  
Vol 8 (12) ◽  
pp. 1277-1280
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
School Work ◽  
New Class ◽  
Class A ◽  
Aluthge Transformation

Research works on Operators in Complex Hilbert spaces has been the interest of budding researchers in the recent years. In 1996, Furuta et al studied Aluthge transformation on phyponormal operators. Later, in 2001 yamazaki et al studied Aluthge transformation and powers of operators for class A(k)operator. This work was further carried over by Pannayappan et al and D.Senthil kumar et al. In this school work, we studied Aluthge transformation and *- Aluthge transformation for the new class of operator named M class Ak * operator on a non-zero Complex Hilbert space.

Uniqueness of Generalized Solutions of Abstract Differential Equations

1975 ◽  
Vol 18 (3) ◽  
pp. 379-382
Author(s):  
M. A. Malik
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Differential Equations ◽  
Open Subset ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Complex Hilbert Space ◽  
Generalized Solutions ◽  
Closed Linear Operator ◽  
Abstract Differential Equations

Let Ω be an open subset of R and H be a complex Hilbert space; (,) represents scalar product in H.Let also A be a closed linear operator with domain DA dense in H and A* with domain D*A be its adjoint. Under graph scalar product DA and D*A are also Hilbert spaces.

scholarly journals A characterization of spectral operators on Hilbert spaces

1982 ◽  
Vol 23 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Ernst Albrecht
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Spectral Operator ◽  
Bounded Linear ◽  
Image Position

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following propertyConversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

scholarly journals Non compact boundaries of complex analytic varieties in Hilbert spaces

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Samuele Mongodi ◽  
Alberto Saracco
Keyword(s):  
Hilbert Space ◽  
Boundary Problem ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Isolated Singularities ◽  
Strongly Convex ◽  
Complex Dimension ◽  
Analytic Varieties ◽  
Complex Varieties ◽  
Compact Case

AbstractWe treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly convex, open subsets of a complex Hilbert space H.We deal with the problem by cutting with a family of complex hyperplanes and applying the already known result for the compact case.

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