توصيفگر ها :
عملگرهاي T-اندازه پذير , نامساوي يانگ , نامساوي هينز , اعداد منفرد , جبر فون نويمن نيمه متناهي , احاطه سازي
چكيده فارسي :
بسياري از نامساوي ها و تساوي ها در ميدان اعداد مختلط داراي تعميم هايي در جبر عملگرها
مي باشد كه اكثراً بر مبناي مفاهيم يكنوايي عملگرها و يا خاصيت تحدب و تقعر آن ها گسترش
مي يابند. اين نوع از نامساوي ها اغلب براي عملگرها روي فضاي هيلبرت متناهي البعد ظاهر
مي شوند. تعداد ديگري از نتايج در مورد نامساوي هاي عملگري نيز از مفاهيم طيف و يا
نامساوي هاي مقادير منفرد حاصل مي شوند. اين مفاهيم مطالعه ي آن دسته از نامساوي ها در
فضاي نامتناهي البعد را ميسر مي نمايد. بر همين اساس در اين نوشتار براي جبر فون نويمن
نيمه متناهي M ،نامساوي هاي يانگ و ميانگين هينز را مورد مطالعه قرار داده و نتايج حاصل
براي عملگرهاي τ-اندازه پذير را ارائه مي دهيم. همچنين بهبود و تظريف اين نامساوي ها را
براي فضاي ناجابه جايي (M(Lp بررسي مي نماييم.
چكيده انگليسي :
Our purpose of this thesis is to formulated some of the norm matrix inequalities at the concept of von Neumann
algebras. The inequalities include forms of the arithmetic-geometric mean, the Cauchy-Schwarz, the Heinz mean inequality, Young’s inequality and some inequalities related to Bourin’s question for τ -measurable operators. Among
many other studies, we focused more on the papers [3, 20] which are regarding extensions to unitary invariant norms
on spaces of matrices. We have also improved and presented a new method for obtaining the inequalities mentioned
in [16, 17]. A number of new inequalities based on [32] are presented. We have obtained the case of equality for
all these inequalities by method used in [15, 29].
The first two chapter of this thesis provide the preliminaries for the discussion of operators inequalities. In chapter
one, the necessary background in operator theory is presented. Such material can be found in an assortment of
texts. Chapter two introduces the aspects of τ -measurable operators, noncommutative measure space (M, τ ) with
a semi finite von Neumann algebra M and faithful normal semi finite trace τ on M.
Let H be an infinite dimensional Hilbert space and let L(H) be the algebra of all bounded operators in H. N is a
von Neumann algebra on H, that is a ∗-subalgebra of L(H) closed in the weak operator topology. The identity in
N is denoted by 1. We are only interested in semi-finite von Neumann algebras, that is, those which admit a faithful
normal semi-finite trace τ . We fix a couple (M, τ ). A von Neumann algebra is said to be σ-finite if it admits at
most countably many orthogonal projections. The cone of positive operators, the identity and the projection lattice
in M are denoted by M+, 1 and P(M) respectively.
An (unbounded) operator x with domain D(x) ⊆ H is densely defined, if D(x) is dense in H. The operator x
is called closed whenever its graph is a closed subspace of H × H. A linear operator x : D(x) → H is called
affiliated with M, if ux = xu for all unitary u ∈ M′
. Note that if x ∈ B(H), then x is affiliated with M if
and only if x ∈ M. A closed and densely defined linear operator x : D(x) → H is said to be τ -measurable
if x affiliated with M, and there exists λ ≥ 0 such that τ (e
|x|
(λ, ∞)) < ∞. In fact, there exists a sequence
{pn}∞
n=1 of orthogonal projections in M such that pn(H) ⊆ D(x) for all n, pn ↑ 1 and τ (1 − pn) ↓ 0 as
n → ∞. The collection of all τ -measurable operators is denoted by S(τ ) or L0(M). The set S(τ ) is a complex
∗-algebra with unit element 1. The von Neumann algebra M is a ∗-subalgebra of S(τ ).
It is known [?] that M is dense in L0(M). In fact, if x = u|x| ∈ L0(M) and |x| =
∫ ∞
0
λde|x|
(λ), then the
sequences {u
∫ n
0
λde|x|
(λ)}∞
n=0 in M tends to x as n → ∞ in the measure topology. Let x be a τ -measurable
operator and t > 0. The t-th singular value of x (or generalized s-numbers) is the number denoted by µt(x) and
for each t ∈ R
+
0
is defined by
µt(x) = inf{∥ xe ∥: e ∈ P(M), τ (1 − e) ≤ t}.
The notation of generalized s-numbers for τ -measurable operators was carefully developed by T. Fack and H.
Kosaki in [13]. In this chapter, for operators x, y ∈ L0(M) we introduce the relation of weak majorization in
symbol x ≺w y if ∫ s
0
µt(y) dt ≤
∫ s
0
µt(x) dt for all s > 0 and majorization in symbol x ≺ y if x ≺w y and
∫ ∞
0
µt(y) dt =
∫ ∞
0
µt(x) dt.
New operators inequalities are presented in the remaining chapters. In the third chapter, Young’s inequalities have
been studied in different forms along with their refinements, and these inequalities have also been studied in the
(weak)majorization form. In the final chapter, Heinz’s inequalities, the weighted Heinz’s inequalities for τ - measurable operators are presented along with their equality cases
