Invertible Operators Research Articles

Hyponormal operators in soft sets

In this paper, we introduce the notion of hyponormal operators in soft Hilbert spaces, some properties of these operators are studied, and in a similar way to [6], a hyponormal soft operator is built from a family of operators. In addition, some results are obtained for self-adjoint, invertible, unitary, unit equivalent and normal soft linear operators that relate the properties of these soft operators with a family of operators in certain Hilbert spaces.

A note on the generalized Drazin–Riesz invertible operators

A note on the generalized Drazin–Riesz invertible operators

B-Weyl and Drazin invertible operators linked by weak pseudo $$S_0$$-demicompactness

B-Weyl and Drazin invertible operators linked by weak pseudo $$S_0$$-demicompactness

Oversmoothing Tikhonov regularization in Banach spaces * *The work of DC was financially supported by National Natural Science Foundation of China (Nos. 11701205 and 11871240). The work of BH was financially supported by German Research Foundation (DFG Grant HO 1454/12-1). The work of IY was financially supported by German Research Foundation (DFG Grants YO159/2-2 and YO159/4-1).

This paper develops a Tikhonov regularization theory for nonlinear ill-posed operator equations in Banach spaces. As the main challenge, we consider the so-called oversmoothing state in the sense that the Tikhonov penalization is not able to capture the true solution regularity and leads to the infinite penalty value in the solution. We establish a vast extension of the Hilbertian convergence theory through the use of invertible sectorial operators from the holomorphic functional calculus and the prominent theory of interpolation scales in Banach spaces. Applications of the proposed theory involving ℓ 1, Bessel potential spaces, and Besov spaces are discussed.

On Lie group representations and operator ranges

In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R \mathcal {R} be an operator range of a Hilbert space H \mathcal {H} . Given the set Λ \Lambda of R \mathcal {R} -invariant operators, and given a Lie group representation ρ : G → GL ( H ) \rho :G\rightarrow \text {GL}(\mathcal {H}) , we discuss the induced semigroup homomorphism ρ ~ : ρ − 1 ( Λ ) → B ( R ) \widetilde {\rho }: \rho ^{-1}(\Lambda ) \rightarrow \mathcal {B(R)} for the operator range topology on R \mathcal {R} . In one direction, we work under the assumption ρ − 1 ( Λ ) = G \rho ^{-1} (\Lambda ) = G , so ρ ~ : G → B ( R ) \widetilde {\rho }:G\rightarrow \mathcal {B}(\mathcal {R}) is in fact a group representation. In this setting, we prove that ρ ~ \widetilde {\rho } is continuous (and smooth) if and only if the tangent map d ρ d\rho is R \mathcal {R} -invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H \mathcal {H} , the set ρ − 1 ( Λ ) \rho ^{-1}(\Lambda ) is neither a group for a large set of nonclosed operator ranges R \mathcal {R} nor closed for all nonclosed operator ranges R \mathcal {R} . Both results are proved by means of explicit counterexamples.

MMIG: Inversion optimization in majority inverter graph with minority operations

Majority inverter graph is a logic representation structure that along with its algebraic properties synthesizes circuits with improved area, delay, and speed metrics, as compared to conventional And-Invert graph (AIG) realization. In this paper, we propose mMIG synthesis approach, where we aim to minimize the number of inverters in such circuits by adopting minority logic in addition to majority and inversion operations in the logic representation. We propose a set of Boolean transformation methods and derived theorems for minority, majority, and inverter operations. We demonstrate that minority operation in addition to majority and inversion operations significantly optimizes the hardware footprint of combinational circuits and cryptographic primitives, such as linear operations and substitution boxes in several lightweight block ciphers. The area optimization is considered with reduction in count of complemented edges or inversion operations. As results demonstrate, the inversion operations have been reduced from 57.7% to 93.3% in mMIG synthesis approach as compared to MIG logic synthesis in EPFL combinational benchmark suite. In round function implementations of lightweight block ciphers such as SIMON, and ARX boxes such as MARX-2 and SPECKEY, the count of complemented edges in mMIG synthesis technique has been reduced by almost 50% as compared to that in MIG based implementations.

The Angle Along a Curve and Range-Kernel Complementarity

We define the angle of a bounded linear operator A along a curve emanating from the origin and use it to characterize range-kernel complementarity. In particular we show that if $$\sigma (A)$$ does not separate 0 from $$\infty $$ , then $$X=R(A)\oplus N(A)$$ if and only if R(A) is closed and some angle of A is less than $$\pi $$ . We first apply this result to invertible operators that have a spectral set that does not separate 0 from $$\infty $$ . Next we extend the notion of angle along a curve to Banach algebras and use it to prove two characterizations of elements in a semisimple and in a $$C^*$$ commutative algebra respectively, whose spectrum does not separate 0 from $$\infty $$ .

Analysis of factors affecting efficiency of inverters: Case study grid-connected PV systems in lower northern region of Thailand

In grid-connected PV systems, the inverter is one of the important components. Inverter efficiency may vary depending on the input power and voltage of the PV array. This paper analysed three factors affecting inverter efficiency. The first one was the effect of the duration of inverter operations. Analysis of the operation of a PV system that has been operating four years showed an annual average inverter efficiency of 0.90, almost equal to the manufacturer’s specification of 0.91. The study showed that there was no significant degradation in inverter efficiency because a cool temperature (of 25°C) was maintained for the inverter storage room through an air conditioning system. The second analysis investigated the effect of the power input from different types of PV module technology. The study showed that the inverter connected to p-Si PV modules operated the highest efficiency at 0.91. However, detailed analyses showed that PV module technology had less or minimal impact on inverter efficiency. It was the power input from the PV module that has influence on the inverter efficiency. The third analysis involved the study of the effect of irradiance distribution on inverter efficiency. The study shows that the inverter operates at the maximum efficiency of 0.90 at irradiance of above 350 W/m2, at which range solar energy potential is at its highest at around 85% of the total generation. This means that inverter converts almost all the energy supplied from solar PV at this irradiance range. In conducting the third analysis, a location-specific equation, ηPHS, for determining inverter efficiency was also developed. The ηPHS equation generates the actual inverter efficiency values based on the local irradiance conditions in Phitsanulok province.

Chaos and frequent hypercyclicity for composition operators

AbstractThe notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Ruzsa in 2015 that for backward weighted shifts on $\ell _p(\mathbb {Z})$, the notions of chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^{p}$-spaces, the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class, an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible operator frequently hypercyclic on $\ell _1$ whose inverse is not frequently hypercyclic is constructed.

B-Fredholm Spectra of Drazin Invertible Operators and Applications

In this article, we consider Drazin invertible operators for study of the relationship between their B-Fredholm spectra and the transfer between some of the spectral properties defined through B-Fredholm spectra of this class of operators. Among other results, we investigate the transfer of generalized a-Weyl’s theorem from T to their Drazin inverse S, if it exists.

Some Multiplicative Inequalities for Heinz Operator Mean

In this paper we obtain some new multiplicative inequalities for Heinz operator mean. 1. Introduction Throughout this paper A; B are positive invertible operators on a complex Hilbert space (H; h ; i) : We use the following notations for operators and 2 [0; 1] Ar B := (1 )A+ B; the weighted operator arithmetic mean, and A] B := A 1=2 A BA 1=2 A; the weighted operator geometric mean [14]. When = 1 2 we write ArB and A]B for brevity, respectively. De ne the Heinz operator mean by H (A;B) := 1 2 (A] B +A]1 B) : The following interpolatory inequality is obvious (1.1) A]B H (A;B) ArB for any 2 [0; 1]: We recall that Specht’s ratio is de ned by [16] (1.2) S (h) := 8> >: h 1 h 1 e ln h 1 h 1 if h 2 (0; 1) [ (1;1) ; 1 if h = 1: It is well known that limh!1 S (h) = 1; S (h) = S 1 h > 1 for h > 0; h 6= 1. The function is decreasing on (0; 1) and increasing on (1;1) : The following result provides an upper and lower bound for the Heinz mean in terms of the operator geometric mean A]B : Theorem 1 (Dragomir, 2015 [6]). Assume that A; B are positive invertible operators and the constants M > m > 0 are such that (1.3) mA B MA: 1991 Mathematics Subject Classi cation. 47A63, 47A30, 15A60, 26D15, 26D10.

Time-dependent non-Hermitian systems: pseudo-squeezed coherent states

We introduce a pseudo-squeezed bosonic ladder operator defined as a time-dependent non-Hermitian linear invariant and related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. In fact, they are obtained from the squeezed transformation of the pseudo-bosons annihilation and creation operators. Thus, the pseudo-bosonic squeezed coherent states are just obtained as pseudo-displacement operator method acting on the ground pseudo-squeezed state. To our knowledge, the time-dependent pseudo-squeezed coherent states have not been constructed until now. As an illustration, we study the time-dependent non-Hermitian displaced harmonic oscillator, and the properties of these states are analysed with respect to the localization in position and to uncertainty principle.

Enhance capability of separable ball criterion for the bipartite quantum states

The largest separable ball is a meaningful geometric sufficient condition for separability of bipartite quantum states which tells that all the states in the unit ball in Frobenius norm centered at the identity matrix are separable. In this paper, we propose an algorithm that can improve the capability of this criterion to detect the separability outside the unit ball by solving an optimization problem based on the invertible local operators, that is, to optimize with respect to the invertible square matrices A and B. The numerical examples have shown that our algorithm is more powerful than the original separable ball criterion.

Orthogonality preserving property for pairs of operators on Hilbert $$C^*$$-modules

We investigate the orthogonality preserving property for pairs of mappings on inner product $C^*$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the $C^*$-valued inner product structure of a Hilbert $C^*$-module is determined essentially by the module structure and by the orthogonality structure, pairs of linear and local orthogonality-preserving mappings are investigated, not a priori bounded. The intuition is that most often $C^*$-linearity and boundedness can be derived from the settings under consideration. In particular, we obtain that if $\mathscr{A}$ is a $C^{*}$-algebra and $T, S:\mathscr{E}\longrightarrow \mathscr{F}$ are two bounded ${\mathscr A}$-linear mappings between full Hilbert $\mathscr{A}$-modules, then $\langle x, y\rangle = 0$ implies $\langle T(x), S(y)\rangle = 0$ for all $x, y\in \mathscr{E}$ if and only if there exists an element $\gamma$ of the center $Z(M({\mathscr A}))$ of the multiplier algebra $M({\mathscr A})$ of ${\mathscr A}$ such that $\langle T(x), S(y)\rangle = \gamma \langle x, y\rangle$ for all $x, y\in \mathscr{E}$. In particular, for adjointable operators $S$ we have $T=(S^*)^{-1}$, and any bounded invertible module operator $T$ may appear. Varying the conditions on the mappings $T$ and $S$ we obtain further affirmative results for local operators and for pairs of a bounded and of an unbounded module operator with bounded inverse, among others. Also, unbounded operators with disjoint ranges are considered. The proving techniques give new insights.

On Subspace-recurrent Operators

In this article, subspace-recurrent operators are presented and it is showed that the set of subspace-transitive operators is a strict subset of the set of subspace-recurrent operators. We demonstrate that despite subspace-transitive operators and subspace-hypercyclic operators, subspace-recurrent operators exist on finite dimensional spaces. We establish that operators that have a dense set of periodic points are subspace-recurrent. Especially, if $T$ is an invertible chaotic or an invertible subspace-chaotic operator, then $T^{n}$, $T^{-n}$ and $\lambda T$ are subspace-recurrent for any positive integer $n$ and any scalar $\lambda$ with absolute value $1$. Also, we state a subspace-recurrence criterion.

Perturbation and construction of almost self‐located robust frames with applications to erasure recovery

In the frame theory, stability is one of the most important properties of frames. In this paper, we investigate the stability of the almost self‐located robust frames, which can reconstruct signals from its unordered partial frame coefficients. We give conditions on the stability of almost self‐located robustness of frames in erasure recovery. We prove that the property of almost self‐located robustness is preserved with respect to an invertible operator, and consequently, every almost self‐robust frame can be transformed into an almost self‐located robust Parseval frame by using a special invertible transformation. We also present some construction methods of almost self‐located robust frames. In particular, we show that every frame can be rescaled to an almost self‐located frame after adding a suitable vector.

Further inequalities involving the weighted geometric operator mean and the Heinz operator mean

ABSTRACT In this paper, we first investigate some inequalities involving the p-weighted geometric operator mean where is a real number and A, B are two positive invertible operators acting on a Hilbert space. As applications, we obtain some inequalities about the so-called Tsallis relative operator entropy. We also give some inequalities involving the Heinz operator mean. Our results refine some inequalities existing in the literature. In a second part, we construct iterative algorithms converging to with a high rate of convergence. Some relationships involving are deduced. Numerical examples illustrating the theoretical results are also discussed.

On Functional Representations of Positive Hilbert Space Operators

In this paper we consider some faithful representations of positive Hilbert space operators on structures of nonnegative real functions defined on the unit sphere of the Hilbert space in question. Those representations turn order relations between positive operators to order relations between real functions. Two of them turn the usual Lowner order between operators to the pointwise order between functions, another two turn the spectral order between operators to the same, pointwise order between functions. We investigate which algebraic operations those representations preserve, hence which kind of algebraic structure the representing functions have. We study the differences among the different representing functions of the same positive operator. Finally, we introduce a new complete metric (which corresponds naturally to two of those representations) on the set of all invertible positive operators and formulate a conjecture concerning the corresponding isometry group.

Triple reverse order law of Drazin invertible operators

In this paper we study the triple reverse-order law (ABC)D = CDBDAD for the Drazin invertible operators A,B and C under the commutative relations [AB, B] = 0, [BC, B] = 0 and [AB, BC] = 0.

Nullspace Property for Optimality of Minimum Frame Angle Under Invertible Linear Operators

Frames with a large minimum angle between any two distinct frame vectors are desirable in many present day applications. For a unit norm frame, the absolute value of the cosine of the minimum frame angle is also known as coherence. Two frames are equivalent if one can be obtained from the other via left action of an invertible linear operator. Frame angles can change under the action of a linear operator. Most of the existing works solve different optimization problems to find an optimal linear operator that maximizes the minimal frame angle (in other words, minimizes the coherence). In the present work, nevertheless, we consider the question: Is it always possible to find an equivalent frame with smaller coherence for a given frame?. In this paper, we derive properties of the initial unit norm frame that can ensure an equivalent frame with strictly larger minimal frame angle compared to the initial one. It turns out that the nullspace property of a certain matrix obtained from the initial frame can guarantee such an equivalent frame. We also present the numerical results that support our theoretical claims.