N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. 2, Frederick Ungar, New York, 1961.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to capture significant signal characteristics which frames provide [22] 23] 24] Frames have been used to design unitary space time constellations for multiple antenna wireless systems [25] Finally, although a well known result by a Russian mathematician M.A. Naimark Naimark s Theorem [26] has been widely used in frame theory in the past few years [8] 27] only recently have researchers recast certain quantum measurement results in terms of frames [28] 29] The bibliography on frames is vast; the list given above is just a sample. The reader is encouraged to check the ....

....latter, we will examine a particular class of frames with vectors which are shifted versions of M prototype ones. This will become clear in a moment. The following theorem tells us that every tight frame can be seen as a projection of an orthonormal basis from a larger space. Theorem 1 (Naimark [26]) A family f i g i2I in a Hilbert space H is a normalized tight frame for H if and only if there is a larger Hilbert space H ae K and an orthonormal basis fe i g i2I for K so that the orthogonal projection P of K onto H satisfies: P e i = i , for all i 2 I . This theorem has been ....

N. I. Akhiezer and I. M. Glazman. Theory of Linear Operators in Hilbert Spaces, volume 1. Frederick Ungar Publisher, 1966.

....hf; gi = Z b a f(x)g(x)w(x) dx; and norm kfk = p hf; fi. We shall say that a function f is square integrable at a if, for some fi 2 (a; b) Z fi a jf(x)j 2 w(x) dx 1: Square integrability at b is defined similarly. Given a function y we define the quasi derivatives y [0] y; y [1] = y 0 ; y [2] py 00 ; y [3] Gamma(py 00 ) 0 sy 0 : 2.3) These quasi derivatives were introduced for scalar 2n th order problems by Naimark [15] see also Everitt and Zettl [8] and Zettl [17] for further information on quasi differential operators. Definition 2.1 The ....

....These quasi derivatives were introduced for scalar 2n th order problems by Naimark [15] see also Everitt and Zettl [8] and Zettl [17] for further information on quasi differential operators. Definition 2. 1 The maximal domain Dmax is the set of functions y whose quasi derivatives y [0] y [1] , y [2] and y [3] are all absolutely continuous, and for which y and y lie in L 2 (a; b; w) The maximal operator Lmax is the operator defined by Lmax y = y on the domain D(Lmax ) Dmax . Definition 2.2 The pre minimal domain Cmin is the set of all functions in Dmax having compact ....

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N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space. Vol. II. Pitman, London (1981).

....It will su#ce to only vaguely describe the full result in this introduction. Definition. A Herglotz function is a function #(z) defined in C # z#C Imz 0 and analytic there with Im #(z) 0there. These are also sometimes called Nevanlinna functions. It is a fundamental result (see, e.g. [2]) that given such a #, there exists c # 0, d real, and a measure d on R with # d(x) 1 x 2 # so that either c # =0ord #=0orboth,and #(z) cz d # # 1 x z x 1 x 2 # d(x) 1.19) THE CLASSICAL MOMENT PROBLEM 13 Theorem 4. The solutions of the Hamburger moment problem ....

....to the structure of M H (#) as a compact convex set. Appendix C summarizes notation and some constructions. 2. The Hamburger Moment Problem as a Self Adjointness Problem Let us begin this section with a brief review of the von Neumann theory of selfadjoint extensions. For further details, see [2, 5, 33, 34]. We start out with a densely defined operator A on a Hilbert space H, that is, D(A) is a dense subset of H and A : D(A) #Ha linear map. We will often consider its graph #(A) #HHgiven by #(A) #,A#) # # D(A) Given operators A, B,wewriteA#Band say B is an extension of A if and ....

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N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. 2, Frederick Ungar, New York, 1961.

....section is to factorize in a suitable way the function z z 1 N(z) in form (0.25) Such a factorization is presented in more general case in [15] The functions b 1m R (#) and 1 b 1 m L (#) are holomorphic in the upper half plane and have positive imaginary part in it. Thus, see, e.g. [14]) taking into account their asymptotic behavior (see (1.1) 1.2) we can represent these functions in the form: b 1m R (#) # # # d#R(# ) # # , 1 b 1m L (#) # b 1 # # # # d#L(# ) # # , 2.3) where, according to the Stieltjes Perron inversion formula, #R(#) ....

N. Akhiezer and I. Glazman, Theory of linear operators in Hilbert spaces. "Nauka", Moskow (1966).

....to capture significant signal characteristics which frames provide [2] 3] 26] Frames have been used to design unitary space time constellations for multiple antenna wireless systems [23] Finally, although a well known result by a Russian mathematician M.A. Naimark 1 Naimark s Theorem [1] has been widely used in frame theory in the past few years [17] 21] only recently have researchers recast certain quantum measurement results in terms of frames [14] 25] The bibliography on frames is vast; the list given above is just a sample. The reader is encouraged to check the ....

....latter, we will examine a particular class of frames with vectors which are shifted versions of M prototype ones. This will become clear in a moment. The following theorem tells us that every tight frame can be seen as a projection of an orthonormal basis from a larger space. Theorem 1 (Naimark [1]) 4 A family f i g i2I in a Hilbert space H is a normalized tight frame for H if and only if there is a larger Hilbert space H ae K and an orthonormal basis fe i g i2I for K so that the orthogonal projection P of K onto H satisfies: P e i = i , for all i 2 I . 3 Actually, the definition of ....

[Article contains additional citation context not shown here]

N. I. Akhiezer and I. M. Glazman. Theory of Linear Operators in Hilbert Spaces, volume 1. Frederick Ungar Publisher, 1966.

....ffl ; Omega Gamma which is indeed compactly embedded in L 2( Omega Gamma1 see [22, 31] Thus, we can show that T maps an H bounded set into an H compact one. Therefore, T : H Gamma H is a compact operator. T is also a compact operator on H as the adjoint of a compact operator, see [1]. It follows from (4.33) that T = T I B, with I B defined in (4.3) solves a (T (E; E 3 ) F ; q) IB(E; E 3 ) F ; q) H 8(E; E 3 ) 2 H 8(F ; q) 2 X: 4.38) Considered on H, T is a compact operator as the product of a bounded operator with a compact one. Using (4.38) we can rewrite ....

N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover, NY, 1993.

....use now the spectral theorem for the self adjoint operator M in H Omega H. Let M f be the spectral measure of f . We have (g; e GammaitM f)H Omega Gamma = Z R e Gammaitx g(x)d M f (x) 17) where g 2 L 2 (R; d M f ) and k gk L 2 (R;d M f ) kgkH Omega Gamma ; 18) see e.g. [1] or [7] The Fourier transform of M f is related to that of in a very simple manner, namely M f (t) f; e GammaitM f)H Omega Gamma = Omega C ; t) Omega C (t) H Omega Gamma = j( t) j 2 = j (t)j 2 : 19) Since for any positive t we have hAi (t) t) A (t) 0, we ....

....fi fi Z g(x)d (x) fi fi fi fi 2 4p d z 2 Z Z jg(x)j 2 jg(y)j 2 d (x)d (y) z 2 (x Gamma y) 2 Gamma p d : The following equalities hold: L(T ) sup g:g2G K(g; T Gamma1 ) LR (T ) sup g:g2GR K(g; T Gamma1 ) Proof. It is well known (see e.g. 13] or [1]) that for any 2 H there exists g 2 L 2 (R; d ) such that k kH = kgk L 2 (R; d ) and ( Z g(x)d (x) d (x) jg(x)j 2 d (x) The converse is also true: for any g 2 L 2 (R; d ) there exists corresponding 2 H . If k k = 1, one can easily verify that L( T ) ....

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I. N. Akhiezer, I.M. Glazman: "Theory of Linear Operators in Hilbert Spaces", Vol II, (F. Ungar Publishing Co.), New-York 1963.

....sort have been studied by R. de L. Kronig and W.G. Penney [17] H.Bethe and R. Peierls [8] L.H. Thomas [20] and others. The first mathematical treatment of such physical systems was given by F. Berezin and L. Faddeev in [7] who used Krein s theory of self adjoint extensions of symmetric operators [1]. Let ffi A is the restriction of the Laplace operator to the domain D( ffi A) fu 2 H 2 : u(0) 0g. Then ffi A is a symmetric operator, the self adjoint extensions of which can be considered as perturbations of Gamma Delta by a zero range (singular) potential supported at the point ....

N. Akhiezer and I.M.Glazman."Theory of Linear Operators in Hilbert Space". Vol. II. Pitman Advanced Publ. Boston, 1981.

....to only vaguely describe the full result in this introduction. Definition. A Herglotz function is a function Phi(z) defined in C j fz 2 C j Im z 0g and analytic there with Im Phi(z) 0 there. These are also sometimes called Nevanlinna functions. It is a fundamental result (see, e.g. [2]) that given such a Phi, there exists c 0, d real, and a measure d on R with R d(x) 1 x 2 1 so that either c 6= 0 or d 6= 0 or both, and Phi(z) cz d Z 1 x Gamma z Gamma x 1 x 2 d(x) 1.19) Theorem 4. The solutions of the Hamburger moment problem in the indeterminate ....

....to the structure of M H (fl) as a compact convex set. Appendix C summarizes notation and some constructions. x2. The Hamburger Moment Problem as a Self Adjointness Problem Let us begin this section with a brief review of the von Neumann theory of selfadjoint extensions. For further details, see [2, 5, 33, 34]. We start out with a densely defined operator A on a Hilbert space H, that is, D(A) is a dense subset of H and 16 B. SIMON A : D(A) H a linear map. We will often consider its graph Gamma(A) ae H Theta H given by Gamma(A) f( A ) j 2 D(A)g. Given operators A; B, we write A ae B and say ....

[Article contains additional citation context not shown here]

N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. 2, Frederick Ungar, New York, 1961.

....integral f(z) Z m(d) Gamma z ; Imz 6= 0: 2.7) Here and below we use integrals without indicated limits denote integrals over whole real axis. f(z) is obviously analytic for non real z and satisfies the conditions Imf Delta Imz 0; Imz 6= 0; sup y1 yjf(iy)j = 1: 2. 8) It can be shown [4] that any function f(z) defined and analytic for non real z and satisfying conditions (2.8) is the Stieltjes transform of a unique nonnegative and normalized to 1 measure m(d) and that for any continuous function ( with a compact support Z ( m(d) lim 0 1 Z ( Imf( i )d (2.9) 4 ....

....in the form j = e i j ; 0 j 2; j = 1; n and to introduce the normalized counting measure (cf. 1.2) N n ( Delta) 1 n f j 2 Deltag (2.43) where Delta is Borel set of [0; 2) An analogue of the Stieltjes transform (2. 7) for measures on the unit circle is the Herglotz transform [4] h(z) Z 2 0 e i z e i Gamma z m(d ) jzj 6= 1: 2.44) Respective inversion formula is (cf. 2.9) Z 2 0 ( m(d ) lim r 1 Gamma0 1 2 Z 2 0 ( Reh(re i )d : We use here instead of (2.18) the differentiation formula E( 0 (M)AM) E( 0 (M)MA) 0 (2.45) valid for any ....

N. Akhiezer, I. Glazman, Theory of linear operators in Hilbert space, Frederick Ungar, New York, 1961.

.... matrix A = a j;k ) j;k0 of complex numbers, can we define correctly a (closed and perhaps densely defined) operator by matrix calculus by identifying elements of 2 with infinite column vectors Of course, since Hilbert and its collaborators, an answer to this question is known, see, e.g. [2]. In this section we briefly summarize the most important facts. Here we will restrict ourselves to matrices A with rows and columns being elements of 2 , an assumption which is obviously true for banded matrices such as our complex Jacobi matrices. By assumption, the formal product A Delta ....

....(2.2) a proper Jacobi matrix A remains proper after adding some bounded perturbation. 2 THE JACOBI OPERATOR 11 If the entries of the difference of two (complex) Jacobi matrices tend to zero along diagonals, then the difference of the corresponding difference operators is known to be compact [2]. We can now give a different characterization of the essential spectrum, namely oe ess (A) foe(A 0 ) A 0 is a difference operator and A Gamma A 0 is compactg: 2.12) Here the inclusion ae is true even in a more general setting [28, Theorem IV.5.35] In order to see the other ....

N.I. Akhiezer & I.M. Glazman, Theory of linear operators in a Hilbert space, volume I,II, Pitman, Boston 1981.

....6j 0, then (4.13) 0: Indeed ( u) satisfies (4. 14) Gammau 00 pu u = 0 where p = ff fi (u)u 2 L 1 (R ) L 1 (R ) The differential operator Lu = Gammau 00 pu has been widely studied; its essential spectrum oe e (L) is [0; 1) see Dunford and Schwartz [1958] Chapter XIII, Akhiezer and Glazman [1981], Naimark [1968] or Schechter [1971] for more details. Let ( 0 ; u 0 ) be a solution of (4.12) with u 0 6j 0 and 0 0. Our goal is to analyze the approximation of solutions of (4.12) in a neighborhood of ( 0 ; u 0 ) making use of the abstract framework in Section 3. The corresponding notations ....

Akhiezer, N.I. and Glazman, I.M., Theory of Linear Operators in Hilbert Spaces, Vol. II, Pitman Advanced Publishing Program, Boston, 1981.

....let us examine the deficiency index of H z . H z is a differential operator of second order with real coefficients and two singular points z = 0; z = 1. Its deficiency index is equal to the number of its orthogonal square integrable eigenfunctions corresponding to a non real eigenvalue (cf. [7]) the deficiency index does not depend on the eigenvalue chosen) In the case s 0 the deficiency index is zero. This comes from the fact that for a fixed non real 2 none of the two linearly independent eigenfunctions I (z) and K (z) is square integrable (cf. Appendix A) In the case s ....

....in this case the deficiency index is one. For s = 0 we have not defined H z as in that case H x is the most suitable form of the Hamiltonian. It is easy to see that the deficiency index of H x is zero when s = 0. To consider the consequences let us start with the case s 0. A theorem of [7] states that if the deficiency index is one then the operator has several self adjoint extensions. Moreover, a condition is given for the different domains of definition of the different self adjoint extensions. The condition has the simplest form if we give it for the H x form of the ....

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N. I. Akhiezer, I. M. Glazman, "Theory of Linear Operators in Hilbert Space", Pitman, 1981.

....L 2 inner product induces an inner product on H(X) we can take the view that H(X) has its own inner product h; i H(X) and likewise its own norm. In our subsequent references to inner products and norms, this view is to be taken. For the general theory of Hilbert space see Akhiezer and Glazman [1] or Riesz and B. Sz. Nagy [31] for the specialization to second order random processes see Rozanov [32] A finite (Q) dimensional second order random field indexed on Z 2 is just a finite collection fX 1 ( m; n) X 2 ( m; n) XQ ( m; n) g of second order random fields indexed on Z 2 . ....

.... from the fact that fU m=M 1 ; m 2 Zg and fU n=N 2 ; n 2 Zg are still groups of unitary operators whose spectral support may be taken as [0; 2 =M) and [0; 2 =N ) To see this, we note the unitary operators U 1=M 1 and U 1=N 2 can be expressed by the spectral theorem for unitary operators [1, 31] as U 1=M 1 = Z 2 0 exp(i 1 =M)dE 1 ( 1 ) 60) and similarly for U 1=N 2 ; then the change of variable fl 1 = 1 =M produces U 1=M 1 = Z 2 =M 0 exp(ifl 1 )dE 1 (M fl 1 ) 61) Harmonizability. The following facts are straightforward extensions of the notion of harmonizable strongly ....

N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Fredrick Ungar, New York, 1961; also published by Dover, New York, 19XX.

.... inclusion from the left turns apparently into equality while the others may be proper (for the most left one to be proper see [4] This suggests the following definition: call a formally normal extension N of S tight if D( S) D(N ) H and tight if D(S ) PD(N ) It has been known ([1] and also [10] that symmetric operators have always selfadjoint (read: normal) extensions which are tight and analytic Toeplitz operator have tight normal extensions as well (cf. 10] while the example of [4] says some subnormal operators do not have any tight normal extension. On the other ....

N.I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert space, vol. I , Pitman, Boston-London-Melbourne. 1981.

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N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. 2, Frederick Ungar, New York, 1961.

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N.I. Akhiezer and I. M. Glazman. Theory of linear operators in Hilbert Spaces. Volume I, Frederick Ungar Publisher, 1966.

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Akhiezer N I and Glazman I M, Theory of Linear Operators in Hilbert Space, Pitman, Boston, 1981.

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N. Akhiezer and I. Glazman. Theory of Linear Operators in Hilbert Space. Ungar, New York, 1963.

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AKHIEZER N.I. and GLAZMAN, I.M. (1981). Theory of linear operators in Hilbert space. I. Pitman, London.

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N. Akhiezer and I. Glazman, Theory of Linear Operators in Hilbert space, vol I, Pitman Publishing, London, 1981.

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N.I. Akhiezer and I. M. Glazman. Theory of linear operators in Hilbert Spaces. Volume I, Frederick Ungar Publisher, 1966.

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N.I. Akhiezer, I.M. Glazman. Theory of Linear Operators in Hilbert Space, Frederick Ungar, Publ., New York, vol. 1, 1966 (translated from Russian by M. Nestell).

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N.I. Akhiezer & I.M. Glazman, Theory of linear operators in a Hilbert space, volume I,II, Nauka, Moscow

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N.I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert space, vol. I, Pitman, Boston-London-Melbourne, 1981.

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