### Citations

30 |
Conditional Measure and Applications,
- Rao
- 1993
(Show Context)
Citation Context ... f 7→ E(f), is called the conditional expectation operator with respect to A. The mapping E is a linear operator and, in particular, it is a contraction operator. In case p = 2, it is the orthogonal projection of L2(Σ) onto L2(A). The role of this operator is important in this note and we list here some of its useful properties: • If f is an A-measurable function, then E(fg) = fE(g). • |E(f)|p ≤ E(|f |p). • If f ≥ 0 then E(f) ≥ 0; if f > 0 then E(f) > 0. • σ(f) ⊆ σ(E(f)), for each nonnegative f ∈ Lp(Σ). • E(|f |2) = |E(f)|2 if and only if f ∈ L0(A). For more details on the properties of E see [11]. Recall that an A-atom of the measure µ is an element A ∈ A with µ(A) > 0 such that for each F ∈ Σ, if F ⊆ A then either µ(F ) = 0 or µ(F ) = µ(A). A measure space (X,Σ, µ) with no atoms 2000 Mathematics Subject Classification. Primary 47B20; Secondary 47B38. Key words and phrases. Conditional multiplication operator, conditional expectation, normal, self-adjoint, closed range, spectrum, compact. 1 2 M. R. JABBARZADEH AND M. R. AZIMI is called non-atomic measure space. It is well-known fact that every σ-finite measure space (X,A, µ|A) can be partitioned uniquely as X = (⋃ n∈NAn ) ∪ B, where {... |

13 |
Weighted conditional expectation operators,
- Herron
- 2011
(Show Context)
Citation Context ...s products involving multiplications and conditional expectations are studied. Some properties and the applications of conditional multiplication operators are also studied in [10] and [7]. In this paper we investigate other operator properties of the members of Kp such as reducibility, closedness of range and compactness. Meanwhile, their kernel and their spectrum are characterized. Finally we close this note off by presenting some examples to illustrate the utility of the results in part. For a beautiful exposition of the study of weighted conditional expectation operators on Lp-spaces, see [4, 5] and the references therein. CONDITIONAL MULTIPLICATION OPERATORS 3 2. Characterization of Conditional Multiplication Operators First note that since R(E) = L2(A) and E is a projection, the Hilbert space L2(Σ) can be decomposed into a direct sum of the subspaces L2(A) and N (E), such that the assignment f 7→ [ E(f) f − E(f) ] is an isometric isomorphism from L2(Σ) onto L2(A) ⊕ N (E). Also note that the matrices of the operators Tu and T ∗ u with respect the above decomposition, denoted by [Tu] and [T ∗ u ], are the following forms (2.1) [Tu] = [ ME(u) 0 Mu−E(u) ME(u) ] and [T ∗u ] = [ M E(u) E... |

12 |
Noncompact composition operators,”
- Singh, Sharma
- 1980
(Show Context)
Citation Context ...w that Sp(Tu) = Sp(T ∗u ) = Sp(ME(u)) = ess rangE(u). Theorem 2.7. Suppose (X,A, µ|A) can be partitioned as X = (⋃ n∈NAn ) ∪ B. Then the bounded linear operator Tu on L p(Σ) is compact if and only if u(x) = 0 for µ-almost all x ∈ B and for any ε > 0, the set {n ∈ N : µ(An ∩Dε(u)) > 0} is finite, where D(u) = {x ∈ X : E(|u|)(x) ≥ ε}. Proof. Recall that Tu is compact if and only if T ∗ u = EMu + ME(u)(I − E) is compact. Since ET ∗u = EMu, so if Tu is compact then EMu is compact. In other hand, if EMu is compact then ME(u) is compact one since EMu|L2(A) = Mu. Thus M E(u) is compact operator([12]). Eventually T ∗u and in turn Tu are compact operators. Consequently, the compactness of Tu is equivalent to the compactness of EMu, which is in turn equivalent to the asserted statement by Theorem 2.2 in [6]. Corollary 2.8. Suppose that (X,Σ, µ) is a non-atomic measure space. Then the bounded linear operator Tu on L 2(Σ) is compact if and only if it is a zero operator. Define Lp = {u ∈ L0(Σ) ∩ D(E) : Tu ∈ Kp}. Then for 1 ≤ p < ∞, u ∈ Lp if and only if E(|u|p) ∈ L∞(A) (see [7]). Hence for u ∈ Lp one can define its norm by ‖u‖Lp := ‖E(|u|p)‖ 1/p ∞ such that (Lp, ‖ · ‖Lp) is respected as a no... |

11 |
Lp multipliers and nested sigma-algebras,
- Lambert
- 1998
(Show Context)
Citation Context ...the following forms (2.1) [Tu] = [ ME(u) 0 Mu−E(u) ME(u) ] and [T ∗u ] = [ M E(u) EM u−E(u) 0 M E(u) ] . In sense of matrix theory, it should be considered that [Tu] is bounded if for each f ∈ L2(Σ), the assignment f 7→ [Tu] [ E(f) f − E(f) ] defines a bounded operator on L2(Σ). A moment’s consideration of Tu’s matrix in (2.1) shows that Tu is a bounded operator if and only if ME(u) : L 2(A) → L2(A), ME(u) : N (E) → N (E) and Mu−E(u) : L2(A) → N (E) are bounded operators. It is known that the boundedness of ME(u) and Mu−E(u) implies that E(u) ∈ L∞(A) and E(|u−E(u)|2) ∈ L∞(A) respectively (see [9]). Since E(|u|2) = E(|u− E(u)|2) + |E(u)|2, it follows that E(|u|2) ∈ L∞(Σ). On the other hand, if E(|u|2) ∈ L∞(A), then E(u) ∈ L∞(A), because |E(u)|2 ≤ E(|u|2). Thus the multiplication operator ME(u) is bounded on the subspaces L2(A) and N (E). Moreover, in this case, we claim that Mu−E(u) is also bounded. Let f ∈ L∞(A) be an arbitrary. Then we have ‖Mu−E(u)f‖22 = ∫ X |u− E(u)|2|f |2dµ = ∫ X E(|u− E(u)|2)|f |2dµ = ∫ X (E(|u|2)− |E(u)|2)|f |2dµ ≤ ‖E(|u|2)‖∞‖f‖22. Then ‖Mu−E(u)‖ ≤ √ ‖E(|u|2)‖∞. Consequently, An operator Tu : L2(Σ) → L2(Σ) is bounded if and only if E(|u|2) ∈ L∞(A). Now, let Tu b... |

10 |
Operators representable as multiplication-conditional expectation operators,
- Grobler, Pagter
- 2002
(Show Context)
Citation Context ... − E(u)E(f) is called the conditional multiplication operator induced by a weight function u. Note that if u is an A-measurable function, then Tu = Mu. Define Kp = {Tu : u ∈ L0(Σ) ∩ D(E) and Tu ∈ B(Lp(Σ))}. For 1 ≤ p <∞, although the Lp(Σ) spaces are not rings, however every member of K∞ has the form Tu ∈ B(L∞(Σ)), u ∈ L∞(Σ). Conditional multiplication operators are closely related to the integral operators (see Example 2.10 (b)), averaging operators on order ideals in Banach lattices and to the operators called conditional expectation-type operators which have been introduced in [1]. Also in [3], operators that are representable as products involving multiplications and conditional expectations are studied. Some properties and the applications of conditional multiplication operators are also studied in [10] and [7]. In this paper we investigate other operator properties of the members of Kp such as reducibility, closedness of range and compactness. Meanwhile, their kernel and their spectrum are characterized. Finally we close this note off by presenting some examples to illustrate the utility of the results in part. For a beautiful exposition of the study of weighted conditional expe... |

2 |
Characterizations of conditional expectation-type operators,
- Dodds, Huijsmans, et al.
- 1990
(Show Context)
Citation Context ...uE(f) + fE(u) − E(u)E(f) is called the conditional multiplication operator induced by a weight function u. Note that if u is an A-measurable function, then Tu = Mu. Define Kp = {Tu : u ∈ L0(Σ) ∩ D(E) and Tu ∈ B(Lp(Σ))}. For 1 ≤ p <∞, although the Lp(Σ) spaces are not rings, however every member of K∞ has the form Tu ∈ B(L∞(Σ)), u ∈ L∞(Σ). Conditional multiplication operators are closely related to the integral operators (see Example 2.10 (b)), averaging operators on order ideals in Banach lattices and to the operators called conditional expectation-type operators which have been introduced in [1]. Also in [3], operators that are representable as products involving multiplications and conditional expectations are studied. Some properties and the applications of conditional multiplication operators are also studied in [10] and [7]. In this paper we investigate other operator properties of the members of Kp such as reducibility, closedness of range and compactness. Meanwhile, their kernel and their spectrum are characterized. Finally we close this note off by presenting some examples to illustrate the utility of the results in part. For a beautiful exposition of the study of weighted con... |

2 |
Lambert multipliers between Lp spaces,
- Jabbarzadeh, Sarbaz
- 2010
(Show Context)
Citation Context ...hough the Lp(Σ) spaces are not rings, however every member of K∞ has the form Tu ∈ B(L∞(Σ)), u ∈ L∞(Σ). Conditional multiplication operators are closely related to the integral operators (see Example 2.10 (b)), averaging operators on order ideals in Banach lattices and to the operators called conditional expectation-type operators which have been introduced in [1]. Also in [3], operators that are representable as products involving multiplications and conditional expectations are studied. Some properties and the applications of conditional multiplication operators are also studied in [10] and [7]. In this paper we investigate other operator properties of the members of Kp such as reducibility, closedness of range and compactness. Meanwhile, their kernel and their spectrum are characterized. Finally we close this note off by presenting some examples to illustrate the utility of the results in part. For a beautiful exposition of the study of weighted conditional expectation operators on Lp-spaces, see [4, 5] and the references therein. CONDITIONAL MULTIPLICATION OPERATORS 3 2. Characterization of Conditional Multiplication Operators First note that since R(E) = L2(A) and E is a projecti... |

2 |
Nagata’s principle of idealization in relation to module homomorphisms and conditional expectations,
- Lambert, Lucas
- 2000
(Show Context)
Citation Context ...p <∞, although the Lp(Σ) spaces are not rings, however every member of K∞ has the form Tu ∈ B(L∞(Σ)), u ∈ L∞(Σ). Conditional multiplication operators are closely related to the integral operators (see Example 2.10 (b)), averaging operators on order ideals in Banach lattices and to the operators called conditional expectation-type operators which have been introduced in [1]. Also in [3], operators that are representable as products involving multiplications and conditional expectations are studied. Some properties and the applications of conditional multiplication operators are also studied in [10] and [7]. In this paper we investigate other operator properties of the members of Kp such as reducibility, closedness of range and compactness. Meanwhile, their kernel and their spectrum are characterized. Finally we close this note off by presenting some examples to illustrate the utility of the results in part. For a beautiful exposition of the study of weighted conditional expectation operators on Lp-spaces, see [4, 5] and the references therein. CONDITIONAL MULTIPLICATION OPERATORS 3 2. Characterization of Conditional Multiplication Operators First note that since R(E) = L2(A) and E is a ... |

1 |
operator matrices,
- Du, Jin
- 1994
(Show Context)
Citation Context ... ≤ 1/δ, and so we get that E(1/|u|) ≤ 1/δ1/p a.e. on σ(E(u)). Therefore, we have |E(g) E(u) χσ(E(u)) |= |E(g)E( 1 u )χσ(E(u))| ≤ |E(g)|E( 1 |u| )χσ(E(u)) ≤ |E(g)| δ 1 p χσ(E(u)). This follows that E(g)/(E(u))χσ(E(u)) ∈ Lp(σ(E(u))). Consequently, E(fn) Lp−→ E(g) E(u) χσ(E(u)) and so fn Lp−→ { g + E(g)− uE(g) E(u) } χσ(E(u)) E(u) := f. Thus Tufn Lp−→ Tuf and hence g = Tuf , which implies that Tu has closed range. Lemma 2.5. Let H and K be separable Hilbert spaces. Suppose that A ∈ B(H), B ∈ B(K) and C ∈ B(K,H). If A and B are normal operators, then Sp ([ A C 0 B ]) = Sp(A) ∪ Sp(B). Proof. See [2]. 6 M. R. JABBARZADEH AND M. R. AZIMI Theorem 2.6. Let u be a nonnegative weight function. Then for a bounded conditional multiplication operator Tu : L 2(Σ)→ L2(Σ) we have Sp(Tu) ∪ {0} = ess rang{E(u)} ∪ {0}. Proof. Let Z = X\σ(E(u)) 6= ∅. Since σ(u) ⊆ σ(E(u)) so by making use of T ∗u ’s matrix, represented in (2.1), we get that L 2(Z,A|Z , µ|Z ) ⊆ N (T ∗u ). Then R(Tu) = N (T ∗u )⊥ ⊆ L2(Zc,A|Zc , µ|Zc ) which means that Tu is not onto and so 0 ∈ Sp(Tu). On the other hand, a moment’s consideration of the matrix of T ∗u and Lemma 2.5 show that Sp(Tu) = Sp(T ∗u ) = Sp(ME(u)) = ess rangE(u). ... |

1 |
Weighted conditional expectation operators on Lp spaces, UNC Charlotte Doctoral Dissertation,
- Herron
- 2004
(Show Context)
Citation Context ...s products involving multiplications and conditional expectations are studied. Some properties and the applications of conditional multiplication operators are also studied in [10] and [7]. In this paper we investigate other operator properties of the members of Kp such as reducibility, closedness of range and compactness. Meanwhile, their kernel and their spectrum are characterized. Finally we close this note off by presenting some examples to illustrate the utility of the results in part. For a beautiful exposition of the study of weighted conditional expectation operators on Lp-spaces, see [4, 5] and the references therein. CONDITIONAL MULTIPLICATION OPERATORS 3 2. Characterization of Conditional Multiplication Operators First note that since R(E) = L2(A) and E is a projection, the Hilbert space L2(Σ) can be decomposed into a direct sum of the subspaces L2(A) and N (E), such that the assignment f 7→ [ E(f) f − E(f) ] is an isometric isomorphism from L2(Σ) onto L2(A) ⊕ N (E). Also note that the matrices of the operators Tu and T ∗ u with respect the above decomposition, denoted by [Tu] and [T ∗ u ], are the following forms (2.1) [Tu] = [ ME(u) 0 Mu−E(u) ME(u) ] and [T ∗u ] = [ M E(u) E... |

1 |
A conditional expectation type operator on Lp spaces,
- Jabbarzadeh
- 2010
(Show Context)
Citation Context ... for µ-almost all x ∈ B and for any ε > 0, the set {n ∈ N : µ(An ∩Dε(u)) > 0} is finite, where D(u) = {x ∈ X : E(|u|)(x) ≥ ε}. Proof. Recall that Tu is compact if and only if T ∗ u = EMu + ME(u)(I − E) is compact. Since ET ∗u = EMu, so if Tu is compact then EMu is compact. In other hand, if EMu is compact then ME(u) is compact one since EMu|L2(A) = Mu. Thus M E(u) is compact operator([12]). Eventually T ∗u and in turn Tu are compact operators. Consequently, the compactness of Tu is equivalent to the compactness of EMu, which is in turn equivalent to the asserted statement by Theorem 2.2 in [6]. Corollary 2.8. Suppose that (X,Σ, µ) is a non-atomic measure space. Then the bounded linear operator Tu on L 2(Σ) is compact if and only if it is a zero operator. Define Lp = {u ∈ L0(Σ) ∩ D(E) : Tu ∈ Kp}. Then for 1 ≤ p < ∞, u ∈ Lp if and only if E(|u|p) ∈ L∞(A) (see [7]). Hence for u ∈ Lp one can define its norm by ‖u‖Lp := ‖E(|u|p)‖ 1/p ∞ such that (Lp, ‖ · ‖Lp) is respected as a normed algebra. Theorem 2.9. (Lp, ‖ · ‖Lp) is a Banach space and for each u ∈ Lp the inequality ‖u‖Lp ≤ ‖Tu‖ ≤ 3‖u‖Lp holds. Proof. First we show that the inequality ‖u‖Lp ≤ ‖Tu‖ ≤ 3‖u‖Lp holds. Suppose u ∈ Lp a... |

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Measurable majorants in L1,
- Lambert
- 1997
(Show Context)
Citation Context ...is σ-finite measure space, we can find a set B ∈ A such that Q := B ∩ σ(E(u)) ⊆ U with 0 < µ(Q) < ∞. Then the A-measurable characteristic function χQ lies in L p(σ(E(u))) and satisfies ‖TuχQ‖pp = ∫ σ(E(u)) |uχQ|pdµ = ∫ σ(E(u)) |u|pχQdµ = ∫ σ(E(u)) E(|u|p)χQdµ ≤ δp ∫ σ(E(u)) χQdµ = δp‖χQ‖pLp(σ(E(u)). This is contrary to the choice of k. Therefore, µ(U) = 0 i.e., E(|u|p) ≥ δ a.e on σ(E(u)). Conversely, suppose E(|u|p) ≥ δ a.e on σ(E(u)) and {Tufn}∞n=0 be an arbitrary sequence in R(Tu) such that ‖Tufn − g‖p → 0 for some g ∈ Lp(Σ). Hence E(Tufn) = E(u)E(fn) Lp−→ E(g). Since, by Proposition 2.9 in [8], E(1/|u|p)χσ(E(u)) = (1/E(|u|p))χσ(E(u)), then we have (E(1/|u|))pχσ(E(u)) ≤ 1/δ, and so we get that E(1/|u|) ≤ 1/δ1/p a.e. on σ(E(u)). Therefore, we have |E(g) E(u) χσ(E(u)) |= |E(g)E( 1 u )χσ(E(u))| ≤ |E(g)|E( 1 |u| )χσ(E(u)) ≤ |E(g)| δ 1 p χσ(E(u)). This follows that E(g)/(E(u))χσ(E(u)) ∈ Lp(σ(E(u))). Consequently, E(fn) Lp−→ E(g) E(u) χσ(E(u)) and so fn Lp−→ { g + E(g)− uE(g) E(u) } χσ(E(u)) E(u) := f. Thus Tufn Lp−→ Tuf and hence g = Tuf , which implies that Tu has closed range. Lemma 2.5. Let H and K be separable Hilbert spaces. Suppose that A ∈ B(H), B ∈ B(K) and C ∈ B(K,H). If A and... |