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ESTIMATES FOR BOUNDS OF AN OPERATOR IN CUT
SUBSPACE OF A FOCK SPACE 1 2 Sharipov I.A. , Rasulov T.H.
1Sharipov Ilhom Azizboy o 'g 'li - Student; 2Rasulov Tulkin Husenovich - Candidate of Physical and Mathematical Sciences,
Head of Department, DEPARTMENT OF MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: in the present paper a 3 x 3 operator matrix A acting in the three- particle cut subspace of a standard Fock space is studied. Analytic description of the essential spectrum of A is obtained. New two-particle and three-particle branches of the essential spectrum of A are singled out. The maximal number of bounded closed intervals of the essential spectrum of A is found. The function (determinant), whose zeroes are coincide with the discrete eigenvalues of A, is defined. Using the cubic numerical the lower and upper bounds of A is estimated.
Keywords: cubic numerical range, operator matrix, standard Fock space, branches of the essential spectrum, bound of the operator.
The numerical range is an important tool in the spectral analysis of bounded and unbounded linear operators in Hilbert spaces. Let H be a complex Hilbert space and let A be a bounded linear operator in H . Then the numerical range of A is the set
W(A) := {(Ax,x):x e SH},
where Sh := {x e H : || x ||= 1} is the unit sphere in H . It was first studied by
O.Toeplitz in 1918 (see [1]); he proved that the numerical range of a matrix contains all its eigenvalues and that its boundary is a convex curve. In 1919 F.Hausdorff showed that
indeed the set W (A) is convex (see [2]). In fact, it turned out that this continues to hold for general bounded linear operators and that the spectrum is contained in the closure W (A) (see [3]).
At the first sight, the convexity of the numerical range seems to be a useful property, e.g. to show that the spectrum of an operator lies in a half plane. However, the numerical range often gives a poor localization of the spectrum and it cannot capture finer structures such as the separation of the spectrum in two parts. In view of these shortcomings, the new concept of quadratic numerical range was introduced in 1998 in [4] and further studied in [5]. If the
Hilbert space H is the product of two Hilbert spaces Hj and H2, that is,
H = H1 0 H2 , then every bounded linear operator A e L(H) has a block operator matrix [6] representation
A =
'Ai
V a21
A
12
A
(1)
22 J
with bounded linear operators Ajj G L(Hj, Hj), i, j = 1,2. In this case the
2
quadratic numerical range ^ (A) of A (with respect to the block operator matrix representation (1)) is the set of all eigenvalues of the 2 X 2 matrices
Af := ((Ajjfj,f j=1, f = (fi,/2)gH, ||fi||= 1, i = 1,2.
The concept of quadratic numerical range for 2 X 2 has an obvious generalization to n X n block operator matrices (see [7,8]), so-called block numerical range. In this paper
we estimate the bounds of a 3 X 3 operator matrix in cut subspace of a Fock space using the cubic numerical range.
First, we give the definition of cubic numerical range. Let H1, H2, H3 be complex Hilbert spaces and consider H = H1 0H2 0H3. With respect to this decomposition, every operator A G L(H) has an 3 X 3 block operator matrix representation
(2)
with bounded linear operators A,j E L(H j, H,), i, j = 1,2,3. In the following
r A11 a12 A13 '
A = a21 a22 a23
v a31 a32 a33 j
we
denote by Sh := Shi x Sh2 x SH3 the product of the unit spheres Sh . in Hi
For f = (fi, /2, /3) E Sh we introduce the 3 X 3 matrix
Af :=((Aj/j f )\ j=1.
Then the set
wh1 eh2 eh3(A):= u (a/)
f esh
is called cubic numerical range of A (with respect to the block operator matrix representation (2)). For a fixed decomposition of H , we also write
w 3( A) = Wh1 e h 2 e h 3( A).
Let H1 := C be the field of complex numbers and Hn := L2([—t; 7r]n — 1) be the
Hilbert space of square integrable (complex) functions defined on [—1 with n = 2,3. Then the Hilbert space H = H1 e H2 e H3 is called three-particle cut subspace of a standard Fock space (truncated Fock space).
Let us consider a 3 X 3 operator matrix A in (1) with the entries
n
Af =f A12/2 = J sin tf2(t)dt, A13 = 0;
-n
A21 = A1*2, (A222f2 )(x) = (s +1- cos x)f2(x) , (A23/3)(x) = J sin t f3(x, t )dt;
-n
A31 = 0, A32=a^ (A-ззfз)(x, y) = (s+2 - cosx - cos y)f3(x, y).
*
Here fi e Hi, i = 1,2,3; s is a real number and Aj is an adjoint operator to Aj .
Under these assumptions the operator A is a bounded and self-adjoint. To study the spectrum of the operator A we introduce a bounded self-adjoint operator (so-called generalized Friedrichs model) Ay which acts in hj 0 h2 as
( A11 42 ^
4 =
V A12 A22 J
According to the Weyl theorem, for the essential spectrum of the operator Ai we have cjess (Ai) = [s; s + 2]. We define an analytic function A(-) (the Fredholm determinant associated with the operator Ai) in C \[s; s + 2] by
... n (sin t )2 dt
A(z) := s - z - J v 7
-ns +1 - cost - z It is easily seen that [9-12]
°disc(A1) = {zeC\[s;s + 2]: A(z) = 0}.
The following theorem describes the essential spectrum of the operator A by the spectrum of A1.
Theorem 1. For the essential spectrum CTess (A) of the operator A the equality
^ (A) = {[0;2] + adiSc (A)} U [s; S + 4]
holds.
Let <7 be the set of complex numbers z e C \[s + 1 - cosx;s + 3 - cosx] such that the equality A(z -1 + cosx) = 0 for some x e [-n; n].
We remark that in [13-15] the essential spectrum of the operator A has been described by zeroes of the Fredholm determinant A( z -1 + cos x) (as a function of z) and the
spectrum of the multiplication operator A33 as follows:
7ess (A) = 7 U [s;s + 4].
We notice that the equality 7 = [0;2] + 7diSc (A1) holds.
Following we introduce the new subsets of the essential spectrum of A . Definition 1. The sets 7 and [s; S + 4] are called two- and three-particle branches of the essential spectrum of A , respectively.
Since the operator A1 has at most 2 simple eigenvalues [16-18], the set C consists no
more than two bounded closed intervals and hence the set Cess (A) consists no more than three bounded closed intervals.
We define an analytic function A(-) in C \ Cess (A) by
* (sin 02 dt Q(z) := e - z - J
A( z -1 + cost)
The following theorem describes the discrete spectrum of the operator A . Theorem 2. The number z G C \ Cess (A) is an eigenvalue of A if and only if Q( z) = 0.
From Theorem 2 it follows that for the discrete spectrum Cdisc (A) of the operator A the equality
Cdrsc (A) = {z G C \ cess (A): Q(z) = 0}
holds. The discrete spectrum of such type operator matrices and its non compact perturbations are studied in many works, see for example [9-12, 19-25].
3
In the following theorem we use the cubic numerical range W (A) to establish
estimates for spectrum of A (see [8, Theorem 7.2]).
Theorem 3. For the lower and upper bounds of A we have
minc(A) = inf W3(A) > e -; maxc(A) = sup W3 (A) < e + 2 + V2*2 + 4 .
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