scholarly journals NEW RESULTS ON SEMICLOSED LINEAR RELATIONS

2021 ◽  
pp. 461
Author(s):  
Gherbi Abdellah ◽  
Messirdi Bekkai ◽  
Messirdi Sanaa
Keyword(s):  
Linear Relation ◽  
Closed Range ◽  
Linear Relations

This paper has triple main objectives. The first objective is an analysis ofsome auxiliary results on closedness and boundednes of linear relations. The seconde objective is to provide some new characterization results on semiclosed linear relations. Here it is shown that the class of semiclosed linear relations is invariant under finite and countable sums, products, and limits. We obtain some fundamental new results as well as a Kato Rellich Theorem for semiclosed linear relations and essentially interesting generalizations. The last objective concern semiclosed linear relation with closed range, where we have particularly established new characterizations of closable linear relation.

scholarly journals On strictly quasi-Fredholm linear relations and semi-B-Fredholm linear relation perturbations

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6337-6355 ◽  
Author(s):  
Bouaniza Hafsa ◽  
Maher Mnif
Keyword(s):  
Finite Rank ◽  
Linear Relation ◽  
Linear Relations ◽  
Finite Rank Operators ◽  
Lower Semi ◽  
The Stability

In this paper we introduce the set of strictly quasi-Fredholm linear relations and we give some of its properties. Furthermore, we study the connection between this set and some classes of linear relations related to the notions of ascent, essentially ascent, descent and essentially descent. The obtained results are used to study the stability of upper semi-B-Fredholm and lower semi-B-Fredholm linear relations under perturbation by finite rank operators.

Left-right fredholm and left-right browder linear relations

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 255-271 ◽  
Author(s):  
T. Álvarez ◽  
Fatma Fakhfakh ◽  
Maher Mnif
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Linear Relation ◽  
Finite Rank Operator ◽  
Linear Relations ◽  
Rank Operator ◽  
Type Decomposition ◽  
Converse Results

In this paper we introduce the notions of left (resp. right) Fredholm and left (resp. right) Browder linear relations. We construct a Kato-type decomposition of such linear relations. The results are then applied to give another decomposition of a left (resp. right) Browder linear relation T in a Banach space as an operator-like sum T = A + B, where A is an injective left (resp. a surjective right) Fredholm linear relation and B is a bounded finite rank operator with certain properties of commutativity. The converse results remain valid with certain conditions of commutativity. As a consequence, we infer the characterization of left (resp. right) Browder spectrum under finite rank operator.

Linear Relations Between Higher Matrix Commutators

1964 ◽  
Vol 16 ◽  
pp. 315-320 ◽  
Author(s):  
Nisar A. Khan
Keyword(s):  
Linear Relation ◽  
Closed Field ◽  
Research Papers ◽  
Algebraically Closed Field ◽  
Linear Relations ◽  
Image Position ◽  
Iterated Commutators

Let Mn denote the space of all n-square matrices over an algebraically closed field F. For A, B ∊ Mn, letdefine the iterated commutators of A and B. Recently several research papers (1, 2, 4, and 5) have appeared on these commutators. In (1), Kato and Taussky have proved that for n = 2 the iterated commutators of A and B satisfy the linear relation

scholarly journals Studies in the dynamics of disinfection

1944 ◽  
Vol 43 (6) ◽  
pp. 363-369 ◽  
Author(s):  
R. C. Jordan ◽  
S. E. Jacobs
Keyword(s):  
Linear Relation ◽  
Death Rate ◽  
Maximum Rate ◽  
Phenol Concentration ◽  
Mortality Level ◽  
Linear Relations ◽  
Low Concentrations ◽  
Constant Rate ◽  
The Times

1. Disinfection curves obtained from data on the action of phenol onBact. coliat 35° C. under conditions such that unfavourable circumstances, other than the presence of the germicide, were as far as possible eliminated, have been used for the calculation of the concentration exponent for phenol, i.e.nin the formulaCnxt=K.The death-rate was not constant throughout the germicidal process but showed initially a phase of slow but increasing death-rate which merged gradually into a phase which was treated (for reasons given) as one of constant rate. This was also the maximum rate for any given phenol concentration.2. The virtual sterilization times (v.s.t.'s), i.e. the times in min. required for the mortality to reach 99·999999 % as determined by slight extrapolation of the log survivors-time curves, the 99·9 % mortality times and the 99 % mortality times could all be used for the calculation of values ofnfor phenol as they all gave satisfactory linear relations between log concentration and log time.3. The 50 % mortality times did not show a satisfactory linear relation between log phenol concentration and log time over the full concentration range, and at this mortality level the concentration exponent appeared to increase for concentrations above 4·62 g. phenol per 1.4. The value ofnvaried according to the mortality level chosen. It was 5·8421 ± 0·1876, 6·6062 ± 0·2034 and 6·9638 ± 0·2164 when thev.s.t.'s,99·9 % mortality times or 99 % mortality times were used. The differences between the first and second and first and third values are significant, but that between the second and third values is not. The value calculated from thev.s.t.'sis regarded as being the most important.5. Evidence was obtained that, as expected on theoretical grounds,nincreases for very low concentrations of phenol. If the aberrant value obtained at the lowest phenol concentration be omitted from the calculations, the value ofncalculated from thev.s.t.'s becomes 5·6588 ± 0·1422, but the decrease is not significant.6. The maximum death-rate was related to the phenol concentration according to the expression km= 9·1743 × 10−6C5·0752, wherekmis the maximum (logarithmic) death-rate per min. andCthe concentration of phenol in g. per 1.

THE EXISTENCE OF ω-CHAINS FOR TRANSITIVE MIXED LINEAR RELATIONS AND ITS APPLICATIONS

2002 ◽  
Vol 13 (06) ◽  
pp. 911-936 ◽  
Author(s):  
ZHE DANG ◽  
OSCAR H. IBARRA
Keyword(s):  
Linear Relation ◽  
Timed Automata ◽  
Unified Framework ◽  
Linear Relations ◽  
Timed Automaton

We show that it is decidable whether a transitive mixed linear relation has an ω-chain. Using this result, we study a number of liveness verification problems for generalized timed automata within a unified framework. More precisely, we prove that (1) the mixed linear liveness problem for a timed automaton with dense clocks, reversal-bounded counters, and a free counter is decidable, and (2) the Presburger liveness problem for a timed automaton with discrete clocks, reversal-bounded counters, and a pushdown stack is decidable.

scholarly journals On the solution of Nevanlinna Pick problem with selfadjoint extensions of symmetric linear relations in Hilbert space

1997 ◽  
Vol 20 (3) ◽  
pp. 457-464
Author(s):  
A. A. El-Sabbagh
Keyword(s):  
Hilbert Space ◽  
Linear Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Resolvent Matrix ◽  
Linear Relations ◽  
Symmetric Linear Relation ◽  
Necessary And Sufficient ◽  
Selfadjoint Extensions ◽  
Pick Problem

The representation of Nevanlinna Pick Problem is well known, see [7], [8] and [11]. The aim of this paper is to find the necessary and sufficient condition for the solution of Nevanlinna Pick Problem and to show that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of symmetric linear relation in Hilbert space. Finally, we define the resolvent matrix which gives the solutions of the Nevanlinna Pick Problem.

scholarly journals Canonical Graph Contractions of Linear Relations on Hilbert Spaces

2021 ◽  
Vol 15 (1) ◽  
Author(s):  
Zsigmond Tarcsay ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Spaces ◽  
Linear Relation ◽  
Regular Part ◽  
Linear Relations ◽  
Closed Linear Relation ◽  
Canonical Graph

AbstractGiven a closed linear relation T between two Hilbert spaces $$\mathcal {H}$$ H and $$\mathcal {K}$$ K , the corresponding first and second coordinate projections $$P_T$$ P T and $$Q_T$$ Q T are both linear contractions from T to $$\mathcal {H}$$ H , and to $$\mathcal {K}$$ K , respectively. In this paper we investigate the features of these graph contractions. We show among other things that $$P_T^{}P_T^*=(I+T^*T)^{-1}$$ P T P T ∗ = ( I + T ∗ T ) - 1 , and that $$Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$$ Q T Q T ∗ = I - ( I + T T ∗ ) - 1 . The ranges $${\text {ran}}P_T^{*}$$ ran P T ∗ and $${\text {ran}}Q_T^{*}$$ ran Q T ∗ are proved to be closely related to the so called ‘regular part’ of T. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed.

Quotient of linear relations and applications

2019 ◽  
Vol 12 (03) ◽  
pp. 1950047
Author(s):  
Abdellah Gherbi ◽  
Bekkai Messirdi
Keyword(s):  
Linear Relation ◽  
Linear Operators ◽  
Topological Properties ◽  
Linear Relations ◽  
Detailed Treatment

In this paper, we extend the notion of quotient of linear operators to linear relations. We introduce for two given linear relations [Formula: see text] and [Formula: see text], the linear relation quotient [Formula: see text] and we give a detailed treatment of some basic algebraic and topological properties of this new notion.

scholarly journals Generalized Kato linear relations

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1129-1139 ◽  
Author(s):  
Mohammed Benharrat ◽  
Teresa Álvarez ◽  
Bekkai Messirdi
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Relation ◽  
Shift Operator ◽  
Linear Operators ◽  
Symmetric Difference ◽  
Linear Relations ◽  
Left Shift ◽  
Kato Spectrum

For a Banach space the notions of generalized Kato linear relation and the corresponding spectrum are introduced and studied. We show that the symmetric difference between the generalized Kato spectrum and the Goldberg spectrum of multivalued linear operators in Banach spaces is at most countable. The obtained results are used to describe the generalized Kato spectrum of the inverse of the left shift operator regarded as a linear relation.

scholarly journals Analytic core and quasi-nilpotent part of linear relations in Banach spaces

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2499-2515
Author(s):  
Maher Mnif ◽  
Aman-Allah Ouled-Hmeda
Keyword(s):  
Banach Spaces ◽  
Linear Relation ◽  
Linear Relations ◽  
Perturbation Result ◽  
Analytic Core

In this paper, we investigate the notion of analytic core and quasi-nilpotent part of a linear relation. Furthermore, we are interested in studying the set of Generalized Kato linear relations to give some of their properties in connection with the analytic core and the quasi-nilpotent part. We finish by giving a perturbation result for this set of linear relations.

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