$$\mu$$ -Norm of an Operator
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Norm of an Operator D. V. Treschev a Received January 17, 2020; revised January 17, 2020; accepted April 8, 2020
Abstract—Let (X , μ) be a measure space. For any measurable set Y ⊂ X let 1Y : X → R be the indicator of Y and let πY be the orthogonal projection L2 (X ) f → πY f = 1Y f . For any bounded operator W on L2 (X , μ) we define its μ-norm W μ = inf χ μ(Yj )W πY 2 , where the infimum is taken over all measurable partitions χ = {Y1 , . . . , YJ } of X . We present some properties of the μ-norm and some computations. Our main motivation is the problem of constructing a quantum entropy. DOI: 10.1134/S008154382005020X
1. INTRODUCTION AND MOTIVATION Let X be a nonempty set and let B be a σ-algebra of subsets X ⊂ X . Consider the measure space (X , B, μ), where μ is a probability measure: μ(X ) = 1. Consider the Hilbert space H = L2 (X , μ) with the scalar product and norm f = f, f . f, g = f g dμ, X
For any bounded operator W on H, let W be its norm defined by W = sup W f . f =1
For any X ∈ B consider the orthogonal projection πX : H → H,
H f → πX f = 1X · f,
(1.1)
where 1X is the indicator of X (the function equal to 1 on X and vanishing at other points). Then μ(X) > 0 implies πX = 1, and for any X , X ∈ B πX + πX = πX ∪X + πX ∩X ,
πX πX = πX ∩X .
We say that χ = {Y1 , . . . , YJ } is a (finite measurable) partition of X if μ X\ Yj = 0, μ(Yj ∩ Yk ) = 0 for any j, k ∈ {1, . . . , J}, k = j. Yj ∈ B, 1≤j≤J
For any two partitions χ = {Y1 , . . . , YJ } and κ = {X1 , . . . , XK }, we define χ ∨ κ = {Yj ∩ Xk }j=1,...,J, k=1,...,K . Obviously, χ ∨ κ is also a partition. a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.
E-mail address: [email protected]
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μ-NORM OF AN OPERATOR
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Let W be a bounded operator on H. For any partition χ = {Y1 , . . . , YJ } of X we define Mχ (W ) =
J
μ(Yj )W πYj 2 .
(1.2)
j=1
We define the μ-norm1 of W by W μ = inf
χ
Mχ (W ).
(1.3)
Recall that the operator U is said to be an isometry if f, g = U f, U g,
f, g ∈ H.
If the isometry U is invertible, then U is called a unitary operator. For any bounded W , any Y ∈ B, and any isometry U , W πY ≤ W ,
U W = W .
This implies the following obvious properties of the μ-norm: W μ ≤ W ,
idμ = 1,
(1.4)
W1 W2 μ ≤ W1 · W2 μ , λW μ = |λ| · W μ W μ = U W μ
(1.5)
for any λ ∈ C,
(1.6)
for any unitary U.
(1.7)
The first question which probably comes to the reader’s mind is “why.” Why such a construction may be useful or interesting? Now we are going to explain our motivations. Let F : X → X be an endomorphism of the measure space (X , B, μ). This means that for any X ∈ B the set F −1 (X) (complete preimage) also lies in B and μ(X) = μ(F −1 (X)). Invertible endomorphisms are called automorphisms. Let End(X ) denote the semigroup of all endomorphisms of (X , B, μ). There are two standard constructions associated with any F ∈ End(X , μ): (1) any such F generates an i
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