Searching for: direct sum of a dilation-type and a von Neumann-type row isometry
Searching for: direct sum of a dilation-type and a von Neumann-type row isometry
Searching for: Lebesgue measure decomposition
Searching for: Lebesgue measure decomposition
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The Lebesgue measure decomposition theorem states that any σ-finite signed measure ν on a measurable space can be written as the sum of two σ-finite signed measures ν0 and ν1, such that ν0 is absolutely continuous with respect to another σ-finite measure μ (denoted by ν0 ≪ μ), and ν1 is singular with respect to μ (denoted by ν1 ⊥ μ). This means that ν0 assigns zero measure to any set that μ assigns zero measure, and ν1 assigns zero measure to any set that has positive μ-measure1
In the case of the real line, the singular part of a regular Borel measure can be further decomposed into three parts: the absolutely continuous part, the singular continuous part, and the pure point part. The absolutely continuous part is the one that has a density function with respect to the Lebesgue measure, the singular continuous part is the one that has no atoms but is supported on a set of Lebesgue measure zero, and the pure point part is the one that consists of atoms (or point masses)1
A row isometry is a linear map from a Hilbert space H to a direct sum of Hilbert spaces H1 ⊕ H2 ⊕ … ⊕ Hn such that it preserves the inner product between any two vectors in H. A row isometry can be represented by a matrix whose rows are isometries from H to Hi. A row isometry is called pure if it has no non-trivial reducing subspaces, absolutely continuous (AC) if it is similar to a direct sum of pure row isometries, von Neumann-type if it is similar to a direct sum of unitary operators, and dilation-type if it is similar to a direct sum of unilateral shifts2
The direct sum of a dilation-type and a von Neumann-type row isometry can be seen as a generalization of the Lebesgue decomposition theorem for measures on the free disk algebra. The free disk algebra is the norm-closed operator algebra generated by the left free shifts, which are isometries on the full Fock space (the space of square-summable power series in several non-commuting variables). The free disk algebra can be viewed as an analogue of the disk algebra in classical Hardy space theory2
The picture shows an illustration of the Lebesgue measure decomposition for a regular Borel measure on the real line. The x-axis represents the real line, and the y-axis represents the measure of each point or interval. The blue curve shows the absolutely continuous part of the measure, which has a density function with respect to the Lebesgue measure. The red dots show the pure point part of the measure, which consists of atoms or point masses. The green curve shows the singular continuous part of the measure, which has no atoms but is supported on a set of Lebesgue measure zero (in this case, the Cantor set). The von Neumann part vanishes smoothly because it corresponds to the singular continuous part, which has no jumps or discontinuities1
chatbot /16.08.2023
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Searching for: light emitting diode
Searching for: light emitting diode
Searching for: cosmological model
Searching for: cosmological model
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That’s an interesting question. A light emitting diode (LED) is a device that converts electrical energy into light by the process of electroluminescence, which is the emission of photons when electrons and holes recombine in a semiconductor material1 A cosmological model is a description of the structure and dynamics of the universe, based on physical laws and observations2
One possible way to make an analogy between an LED and a cosmological model is to compare the different components and properties of each system. For example:
- The electrical current that flows through an LED can be compared to the expansion of the universe, which is driven by the energy density and pressure of its components, such as dark energy, dark matter, and ordinary matter3
- The semiconductor material that forms the core of an LED can be compared to the space-time fabric of the universe, which is described by the theory of general relativity and can be curved by the presence of matter and energy2
- The color of the light emitted by an LED depends on the energy gap between the valence and conduction bands of the semiconductor material, which determines the wavelength of the photons produced1 Similarly, the temperature of the cosmic microwave background (CMB) radiation, which is the remnant of the primordial light from the early universe, depends on the redshift factor that stretches or compresses its wavelength as it travels across the expanding space2
- The brightness or intensity of the light emitted by an LED depends on the amount and direction of the current that flows through it, as well as the efficiency of the electroluminescence process1 Likewise, the luminosity or brightness of distant galaxies and supernovae depends on their intrinsic properties, such as mass, size, and age, as well as their distance and orientation relative to us2
These are just some examples of how one can use an analogy to understand or explain complex concepts in physics and cosmology. However, one should also be aware of the limitations and pitfalls of using analogies, as they can sometimes oversimplify or distort reality. Analogies are useful tools for learning and communication, but they are not substitutes for rigorous mathematical models and empirical evidence4
I hope this helps you with your analogy. If you have any other questions or comments, please feel free to ask me. 😊
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Following is a research paper in which for that conclusion i sought above answers by BING/26.08.23
LEBESGUE DECOMPOSITION OF NON-COMMUTATIVE MEASURES MICHAEL T. JURY AND ROBERT T.W. MARTIN Abstract. The Riesz-Markov theorem identifies any positive, finite, and regular Borel measure on the complex unit circle with a positive linear functional on the continuous functions. By the Weierstrass approximation theorem, the continuous functions are obtained as the norm closure of the Disk Algebra and its conjugates. Here, the Disk Algebra can be viewed as the unital norm-closed operator algebra of the shift operator on the Hardy Space, H2 of the disk. Replacing square-summable Taylor series indexed by the non-negative integers, i.e. H2 of the disk, with square-summable power series indexed by the free (universal) monoid on d generators, we show that the concepts of absolutely continuity and singularity of measures, Lebesgue Decomposition and related results have faithful extensions to the setting of ‘noncommutative measures’ defined as positive linear functionals on a non-commutative multivariable ‘Disk Algebra’ and its conjugates. 1. Introduction The results of this paper extend the Lebesgue decomposition of any finite, positive and regular Borel measure, with respect to Lebesgue measure on the complex unit circle, from one to several non-commuting (NC) variables. In [1], we extended the concepts of absolute continuity and singularity of positive measures with respect to Lebesgue measure, the Lebesgue Decomposition, and the Radon-Nikodym formula of Fatou’s Theorem to the non-commutative, multi-variable setting of ‘NC measures’, i.e. positive linear functionals on a certain operator system, the Free Disk System. Here, the free disk system, Ad + A∗ d , is the operator system of the Free Disk Algebra, Ad := Alg(I, L) −k·k, the norm-closed operator algebra generated by the left free shifts on the NC Hardy space. (Equivalently, the left creation operators on the full Fock space over C d .) We will recall in some detail below why this is the appropriate (and even canonical) extension of the concept of a positive measure on the circle to several non-commuting variables. The primary goal of this paper is to further develop the NC Lebesgue Decomposition Theory of an arbitrary (positive) NC measure with respect to NC Lebesgue measure (the ‘vacuum state’ on the Fock space), by proving that our concepts of absolutely continuous (AC) and singular NC measures define positive hereditary cones, and hence that the Lebesgue Decomposition commutes with summation. That is, the Lebesgue Decomposition of the sum of any two NC measures is the sum of the Lebesgue Decompositions. (Here, we say a positive cone, P0 ⊂ P is hereditary in a larger positive cone P if p0 ∈ P0, and p0 ≥ p for any p ∈ P implies that p ∈ P0. The sets of absolutely continuous and First named author partially supported by NSF grant DMS-1900364. 1 singular positive, finite, regular Borel measures on the circle, ∂D, with respect to another fixed positive measure, are clearly positive hereditary sub-cones.) In this paper we focus on positive NC measures and their Lebesgue decomposition with respect to NC Lebesgue measure. The study of more general complex NC measures, and the Lebesgue Decomposition of an arbitrary positive NC measure with respect to another will be the subject of future research. By the Riesz-Markov Theorem, any finite positive Borel measure, µ, on ∂D, can be identified with a positive linear functional, ˆµ on C (∂D), the commutative C ∗−algebra of continuous functions on the circle. By the Weierstrass Approximation Theorem, C (∂D) = (A(D) + A(D) ∗ ) −k·k∞ , where A(D) is the Disk Algebra, the algebra of all analytic functions in the complex unit disk, D, with continuous extensions to the boundary. In the above formula, elements of A(D) are identified with their continuous boundary values and k · k∞ denotes the supremum norm for continuous functions on the circle. The disk algebra can also be viewed as the norm-closed unital operator algebra generated by the shift, S := Mz, A(D) = Alg(I, S) −k·k (with equality of norms). The shift is the isometry of multiplication by z on the Hardy space, H2 (D), and plays a central role in the theory of Hardy spaces. Here recall that the Hardy Space, H2 (D), is the space of all analytic functions in D with square-summable MacLaurin series coefficients (and with the ℓ 2 inner product of these Taylor series coefficients at 0 ∈ D). The positive linear functional ˆµ is then completely determined by the moments of the measure µ: µˆ(S k ) := Z ∂D ζ kµ(dζ). The shift on H2 (D) is isomorphic to the unilateral shift on ℓ 2 (N0), where N0, the non-negative integers, is the universal monoid on one generator. A canonical several-variable extension of ℓ 2 (N0) is then ℓ 2 (F d ), where F d is the free (and universal) monoid on d generators, the set of all words in d letters. One can define a natural d−tuple of isometries on ℓ 2 (F d ), the left free shifts, Lk, 1 ≤ k ≤ d defined by Lkeα = ekα where α ∈ F d and {eα} is the standard orthonormal basis. These left free shifts have pairwise orthogonal ranges so that the row operator L := (L1, ..., Ld) : ℓ 2 (F d ) ⊗ C d → ℓ 2 (F d ) is an isometry from d copies of ℓ 2 (F d ) into one copy which we call the left free shift. This Hilbert space of free square-summable sequences can also be identified with a ‘NC Hardy Space’ of ‘non-commutative analytic functions’ in a non-commutative open unit disk or ball of several matrix variables. Under this identification, the left free shifts become left multiplication by independent matrix-variables, see Section 2. The immediate analogue of a positive measure in this non-commutative (NC) multi-variable setting is then a positive linear functional, or NC measure, on the Free Disk System: (Ad + A ∗ d ) −k·k , where Ad := Alg(I, L) −k·k is the Free Disk Algebra, the operator norm-closed unital operator algebra generated by the left free shifts. As in the classical theory, elements of the Free Disk Algebra can be identified with bounded (matrix-valued) analytic functions (in several non-commuting matrix variables) which extend continuously from the interior to the boundary of a certain non-commutative multi-variable open unit disk or ball. 2 There is a fundamental connection between this work and the theory of row isometries (isometries from several copies of a Hilbert space into itself), or equivalently to the representation theory of the Cuntz-Toeplitz C ∗−algebra, Ed = C ∗ (I, L) (the C ∗− algebra generated by the left free shifts), the universal C ∗−algebra generated by a d−tuple of isometries with pairwise orthogonal ranges, and of the Cuntz C ∗−algebra, Od, the universal C ∗−algebra of an onto row isometry [2]. Namely, applying the Gelfand-Naimark-Segal (GNS) construction to (µ, Ad), where µ is any (positive) NC measure yields a GNS Hilbert space, F 2 d (µ), and a ∗−representation πµ of Ed so that Πµ := πµ(L) is a row-isometry on F 2 d (µ). A Lebesgue Decomposition for bounded linear functionals on the Free Disk Algebra, Ad, has been developed by Davidson, Li and Pitts in the theory of Free Semigroup Algebras, i.e. W OT−closed (weak operator topology closed) operator algebras generated by row isometries [3, 4, 5]. Building on this, Kennedy has constructed a Lebesgue Decomposition for row isometries [6], and we will explicitly work out the relationship between this theory and our Lebesgue Decomposition. 1.1. Three approaches to classical Lebesgue Decomposition Theory. There are three approaches to classical Lebesgue Decomposition theory which will provide natural and equivalent extensions to NC measures. Let µ be an arbitrary finite, positive, and regular Borel measure on ∂D, and as before, m denotes normalized Lebesgue measure on the circle. As we will prove, one can construct the Lebesgue decomposition of µ with respect to m using reproducing kernel Hilbert space theory. Namely, setting H2 (µ) to be the closure of the analytic polynomials in L 2 (µ, ∂D), let H +(Hµ) be the space of all Cauchy Transforms of elements in H2 (µ): If h ∈ H2 (µ), (Cµh)(z) := Z ∂D 1 1 − zζ h(ζ)µ(dζ). Equipped with the inner product of H2 (µ), this is a reproducing kernel Hilbert space of analytic functions in D, the classical Herglotz Space with reproducing kernel: Kµ (z, w) = 1 2 Hµ(z) + Hµ(w) ∗ 1 − zw∗ = Z ∂D 1 1 − zζ 1 1 − ζw µ(dζ), and Hµ(z) := Z ∂D 1 + zζ 1 − zζ h(ζ)µ(dζ) = 2(Cµ1)(z) − µ(∂D), is the Riesz-Herglotz integral transform of µ, an analytic function with non-negative real part in D (see [7, Chapter 1], or [8, Chapter 1, Section 5]). It is not hard to verify that domination of finite, positive, and regular Borel measures is equivalent to domination of the Herglotz kernels for their reproducing kernel Hilbert spaces of Cauchy Transforms: 0 ≤ µ ≤ t 2λ ⇔ Kµ ≤ t 2Kλ ; t > 0. 3 Moreover, by a classical result of Aronszajn, domination of the reproducing kernels Kµ ≤ t 2Kλ is equivalent to bounded containment of the corresponding Herglotz spaces on D, H +(Hµ) ⊆ H +(Hλ), and the least such t > 0 is the norm of the embedding map eµ : H +(Hµ) ֒→ H +(Hλ) [9, Theorem I, Section 7]. Absolute continuity of measures on ∂D can also be recast in terms of containment of reproducing kernel Hilbert spaces. Namely, given two finite, positive, regular Borel measures λ, µ, recall that µ is absolutely continuous with respect to λ if there is a non-decreasing sequence of finite, positive, regular Borel measures µn, which are each dominated by λ, and increase monotonically to µ: 0 ≤ µn ≤ µ, µn ↑ µ, µn ≤ t 2 nλ, tn > 0. Reproducing kernel Hilbert space theory then implies that each space of µn−Cauchy transforms is contractively contained in the space of µ−Cauchy transforms, and their linear span is dense in H +(Hµ) since the µn increase to µ, _ H +(Hµn ) = H +(Hµ). Since each µn ≤ t 2 nλ, is dominated by λ, it also follows that each space of µn−Cauchy transforms is boundedly contained in the space of λ−Cauchy transforms, and the intersection space: int(µ, λ) := H +(Hµ) \ H +(Hλ), is dense in the space of µ−Cauchy transforms. In the case where λ = m is normalized Lebesgue measure, one can check that Hm ≡ 1 is constant, so that H +(Hm) = H2 (D) is the classical Hardy space of the disk. It follows that one can take this as a starting point, and simply define a measure, µ, to be absolutely continuous or singular (with respect to m) depending on whether the intersection space int(µ, m) := H +(Hµ) \ H2 (D), is dense or trivial, respectively, in the space of µ−Cauchy Transforms. In this way one can develop Lebesgue Decomposition Theory using reproducing kernel techniques. It appears that this approach is new, even in this classical setting, and as shown in Corollary 8.5, this recovers the Lebesgue decomposition of any finite, positive and regular Borel measure on the unit circle with respect to normalized Lebesgue measure. As discussed in the introduction, any positive, finite, regular Borel measure, µ, on ∂D, can be viewed as a positive linear functional, ˆµ, on A(D) + A(D) ∗ . Equivalently, µ (or ˆµ) can be identified with the (generally unbounded) positive quadratic or sesquilinear form, qµ(a1, a2) := Z ∂D a1(ζ)a2(ζ)µ(dζ); a1, a2 ∈ A(D), densely-defined in H2 (D). Applying the theory of Lebesgue Decomposition of quadratic forms due to B. Simon yields: qµ = qac + qs, 4 where qac is the maximal positive form absolutely continuous to qm, where m is normalized Lebesgue measure, and qs is singular [10]. In this theory, a positive quadratic form with dense domain in a Hilbert space, H, is said to be absolutely continuous if it is closable, i.e. it has a closed extension. Here, a positive semi-definite quadratic form, q, is closed if its domain, Dom(q) is complete in the norm k · kq+1 := p q(·, ·) + h·, ·iH. Closed positive semi-definite forms obey an extension of the Riesz Representation Lemma: A positive semi-definite densely-defined quadratic form, q, is closed if and only if q is the quadratic form of a closed, densely-defined, positive semi-definite operator, T ≥ 0: q(h, g) = h √ T h, √ T giH; h, g ∈ Dom(√ T) = Dom(q), see [11, Chapter VI, Theorem 2.21, Theorem 2.23]. If q = qµ, we will prove that qac = qµac , and qs = qµs where µ = µac + µs, is the classical Lebesgue Decomposition of µ with respect to m, see Corollary 8.5. Indeed, if one instead defines qµ as a quadratic form densely-defined in L 2 (∂D), then it follows without difficulty in this case that T is affiliated to L∞(∂D) so that qµ(f, g) = Z ∂D f(ζ)g(ζ)|h(ζ)| 2m(dζ), where √ T1 = |h| ∈ L 2 (∂D). This shows that |h| 2 ∈ L 1 (∂D) is the Radon-Nikodym derivative of µ with respect to normalized Lebesgue measure, m. The Lebesgue Decomposition of quadratic forms in [10] is similar in this case to von Neumann’s proof of the Lebesgue Decomposition theory [12, Lemma 3.2.3]. In [1], we applied this quadratic form decomposition to the quadratic form, qµ, of any (positive) NC measure µ to construct an NC Lebesgue decomposition of µ, µ = µac + µs into absolutely continuous and singular NC measures µac and µs, 0 ≤ µac, µs ≤ µ, where qµ = qµac +qµs is the Lebesgue decomposition of the quadratic form qµ [1, Theorem 5.9]. A third approach to Lebesgue Decompositon theory is to define a positive, finite, regular, Borel measure µ, on ∂D to be absolutely continuous if the corresponding linear functional ˆµ on C (∂D) = (A(D) + A(D) ∗ ) −k·k has a weak−∗ continuous extension to a linear functional on (H∞(D) + H∞(D) ∗ ) −wk−∗ = L∞(∂D). This notion of absolute continuity for bounded linear functionals on Ad extends the classical notion of absolute continuity of a measure with respect to normalized Lebesgue measure on ∂D, if one identifies finite positive Borel measures on ∂D with positive linear functionals on the classical Disk Algebra A1 = A(D) ⊂ H∞(D). Indeed in the case where d = 1, L∞ 1 = H∞(D), and (H ∞(D) + H ∞(D) ∗ ) −weak−∗ ≃ L ∞(∂D), a commutative von Neumann algebra. In this case, if ˆµ ∈ (A(D) † )+ = C (∂D) † + is any positive linear functional, the Riesz-Markov-Kakutani Theorem implies it is given by integration against a positive finite Borel measure, µ, on ∂D, and to say it has a weak−∗ continuous extension to (H∞(D) + 5 H∞(D) ∗ ) † + ≃ L∞(∂D) † + is equivalent to ˆµ being the restriction of a positive ˆµ ∈ L∞(∂D)∗ ≃ L 1 (∂D). Equivalently, µ(dζ) = µ(dζ) m(dζ) m(dζ); m − a.e., µ(dζ) m(dζ) ∈ L 1 (∂D), i.e. µ is absolutely continuous with respect to Lebesgue measure. This definition of absolute continuity has an obvious generalization to the non-commutative setting of NC measures, i.e. positive linear functionals on the Free Disk System, and this gives essentially the same definition of absolute continuity for linear functionals on the Free Disk Algebra introduced by Davidson-Li-Pitts [3]. We will show that all three of these approaches extend naturally to the NC setting and yield the same Lebesgue Decomposition of any positive NC measure with respect to NC Lebesgue measure.
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