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A Review Note on the Applications of Linear Operators in Hilbert Space

Written By

Karthic Mohan and Jananeeswari Narayanamoorthy

Submitted: March 20th, 2020 Reviewed: May 8th, 2020 Published: April 7th, 2021

DOI: 10.5772/intechopen.92773

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Abstract

Hilbert Spaces are the closest generalization to infinite dimensional spaces of the Euclidean Spaces. We Consider Linear transformations defined in a normed space and we see that all of them are Continuous if the Space is finite Dimensional Hilbert Space Provide a user-friendly framework for the study of a wide range of subjects from Fourier Analysis to Quantum Mechanics. The adjoint of an Operator is defined and the basic properties of the adjoint operation are established. This allows the introduction of self Adjoint Operators are the subject of the section.

Keywords

  • linear space
  • norm of a vector
  • inner product
  • orthogonal vector
  • adjoint of an operator

1. Introduction

The Concept of Hilbert Space was put forwarded by David Hilbert in his work on Quadratic forms in infinitely many Variables. We take a Closer look at Linear Continuous map between Hilbert Spaces [1]. These are called bounded operators and branch of Functional Analysis Called “Operator Theory” [2]. Next we derive an important inequality which has many interesting applications in the theory of inner product spaces and as a consequence we obtain that each inner product space is a normed Vector spaces with the norm [3], i.e. the inner product generates this form. Moreover there are several essential algebraic identities, variously and ambiguously called Polarization Identities. These and other closely related identities are of constant use. Now we are in position to state and prove the above mentioned important inequality known as Cauchy-Schwartz Buniakowsi inequality (briefly we say CSB inequality) and we shall also use this to define the concept of angle by means of a formula [1]. The theory of Hilbert Space that Hilbert and Others developed has not only greatly enriched the world of Mathematics but has Proven Extremely useful in the development of Scientific Theories Particularly Quantum Mechanics.

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2. Definition 1

A Hilbert space is a complex Banach space whose norm induced from an inner produce [4] i.e., in which there is defined a complex function xyof vectors x& yand α,βare scalars with the following properties

  1. αx+βyz=αxz+βyz

  2. x,y¯=yx

  3. xx=x2

2.1 Remark 1.1

Every polynomial equation of the nthdegree with complex co-efficient has exactly ncomplex roots [5].

In accordance with the above remarks the scalars in this example are understood to be the complex number.

Consider the space l2nwith the inner product of two vectors x=x1x2xnand y=y1y2yndefined by

xy=i=1nxyy¯i

Now we are going to prove that l2nis a Hilbert space.

2.1.1 Proof

By using Hilbert Space definition [xyof complex function, α,βare scalars]

  1. αx+βyz=i=1nαxi+βyiz¯i

    =αi=1nxiz¯i+βi=1nyiz¯i

    αx+βyz=αxz+βyzx,y,zl2n

  2. x,y¯=i=1nxiy¯i¯

    =i=1nx¯iy¯¯i

    =i=1nyix¯i

    =yx

  3. xx=i=1nxix¯i

    =i=1nxi2

    xx=x2

Therefore xyis an inner product on l2n.

Therefore l2nis a Hilbert space.

2.2 Theorem 1.1 (Schwartz inequality)

If xand yare any two vectors in Hilbert space then xyxy[6, 7, 8].

2.2.1 Proof

When y=0, the result is clear for both sides vanish

i.e., xy=x0=0=0

xy=x0=x0=0
xy=xy=0

When y0

Take any scalar αC[Complex banach space] always xαy20

xαyxαy0
xxxαyαyx+αyαy0
xxα¯xyαyx+αα¯yy0
x2α¯xyαxy¯+αα¯y20E1

Put α=xyy2where xyC.

 y0and y0.

So choose α=xyy2,α¯=x,y¯y2.

From Eq. (1) becomes

x2xy¯y2xyxyy2xy¯+xyy2xy¯y2y20
x2xy2y2xy2y2+xy2y20
x2xy2y20
xy2y2x2
xy2x2y2
xyxy

2.3 Result 1

An inner product space is a normal linear space [9].

2.3.1 Proof

To prove.

x0and x=0if x=0

x=xxx2=xx0

So that x0and x=0x=0.

Now we have to show that x+yx+y

x+y2=x+yx+y
=xx+xy+yx+yy
=x2+xy+xy¯+y2
=x2+2Rexy+y2
x2+2xy+y2
x+y2x+y2
x+yx+y

Now we can prove that αx=αx.

Consider

αx2=αxαx
=αα¯xx
αx2=α2x2
αx=αx

An inner product is a normed linear space.

2.4 Result 1.1

An inner product in Hilbert space is jointly continuous [10].

2.4.1 Proof

Since xnxand ynyxnynxy

We have.

xnx0as nand

yny0as nand

Now consider

xnyn(xy)=xnyn(xny)+(xny)(xy)
xnyn(xny)+xny(xy)

xnyny+xnxy(by Schwartz Inequality)

0as ynyand xnx

xnyn(xy)0as n

xnynxyas n

2.5 Theorem 1.1 (parallelogram law in Hilbert space)

If xand yare any two vectors in Hilbert space then

x+y2+xy2=2x2+2y2E2

2.5.1 Proof

x+y2=x+yx+y
=xx+xy+yx+yy
=x2+xy+xy¯+y2E3
xy2=xyxy
=xxxyyxyy
=x2xyxy¯+y2E4

Adding (3) and (4) we get,

x+y2+xy2=2x2+2y2

2.6 Theorem 1.2 (polarization identity)

If xand yare any two vectors in Hilbert space then

4xy=x+y2xy2+ix+iy2ix+iy2E5

2.6.1 Proof

x+y2=x+yx+y
=xx+xy+yx+yy
=x2+xy+xy¯+y2E6
xy2=xyxy
=xxxyyx+yy
=x2xyxy¯+y2E7

Subtracting (6) and (7) we get

x+y2xy2=2xy+2yxE8

Replace yby iyin Eq. (8)

x+y2xy2=2xiy+2iyx
=2i¯xy+2iyx
=2ixy+2iyx

Multiply both sides by i

ix+y2ixy2=2xy2yxE9

Adding (8) and (9) we get

x+y2xy2+ix+y2ixy2=4xy.E10
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3. Definition 2

Let Bbe an arbitrary banach space. A convex set in Bis a non-empty subset Swith the property that if xand yare in Sthen

z=x+tyx=1tx+ty

it also in Sfor all real number t such that 0t1.

A convex set is a non-empty set which contains the segment joining any pairs of its points.

Since Cis convex it is non-empty and contains x+y2whenever it contains xand y[11].

3.1 Theorem 2.1

3.1.1 Application of parallelogram law

A closed convex subset C of a Hilbert space Hcontains a unit vector of smallest norm [12].

3.1.2 Proof

Step 1:

Since Cis a convex set, it is non empty and contains x+y2Cwhenever x,yC.

Let d=infx/xCthen

dxxCE11

Then there exist a sequence xnof vectors in Csuch that xndas

nE12

Let xm,xnC.

Cis convex, xm+xn2C

By (11), dxm+xn2

dxm+xn2
2dxm+xn
4d2xm+xn2
4d2xm+xn2E13

By parallelogram law

xm+xn2+xmxn2=2xm2+2xn2
xmxn2=2xm2+2xn2xm+xn2

xmxn22xm2+2xn24d2by Eq. (13)

2d2+2d24d2by Eq. (12) as n.

xmxn0as n, m.

xnis a Cauchy sequence in C.

Cis a closed set in complete Space H

Cis complete

There exist a vector xin Csuch that xnx

i.e.) x=limxn

x=limxn
=limxn=d
=infxxC

xis smallest.

xis a vector in Cwith smallest norm.

Step 2:

To prove uniqueness of x.

Suppose there exist a vector xin Cwith x=dand xxin C

Cis convex, x+x2C

By Eq. (11)

dx+x2E14

By parallelogram law,

x+x22+xx22=2x22+2x22
x+x22=2x22+2x22x+x22
2x22+2x22
2x22+2x22
2d24+2d24
4d24
x+x22d2
x+x2d

Which is a contradiction to Eq. (14)

Therefore our assumption on xxis wrong

Hence x=x

3.2 Theorem 2.2 (orthogonal complements)

Two Vectors xand yin a Hilbert space Hare said to be orthogonal (written as xy) if xy=0[9]

  1. xyxy=0

    x,y¯=0¯

    yx=0

    yx

  2. 0x=0xϵH

    0xxϵH

    Therefore 0is orthogonal to every vector xin H

  3. xxxx=0

x2=0
x=0
x=0

This means that 0is the only vector orthogonal to itself

3.3 Theorem 2.3 (Pythagorean theorem)

Geometric fact about orthogonal vectors in the Pythagorean theorem such that xyimplies [9]

x+y2=xy2=x2+y2

3.3.1 Proof

Since

xyxy=0
x,y¯=0¯
yx=0
yx
x+y2=x+yx+y
=xx+xy+yx+yy

=x2+y2by

xy=0,yx=0E15
xy2=xyxy
=xxxyyx+yy

=x2+y2by

xy=0,yx=0E16

From Eq. (15) and (16)

x+y2=xy2=x2+y2
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4. Definition 3

A Vector xis said to be orthogonal to a non-empty set S(written as xs) if xyyϵS[7].

4.1 Definition 3.1

The orthogonal complement of Sdenoted by Sis the set of all vectors orthogonal to si.e., S=x/xHandxs.

i.e., xSxs[10].

The following statements are the easy consequence of the definition

  1. 0=H

  2. H=0

  3. SS0

  4. S1S2S2S1

  5. Sis a closed subspace of H.

4.1.1 Theorem 3.1.1

If Mis a proper closed linear subspace of a Hilbert Space H. Then there exists a non-zero vectors z0in Hsuch that orthogonal to M. i.e., z0M[10].

4.1.1.1 Proof

Let xbe a vector not in Mand let dbe the distance from rto M. Then by theorem “Let Mbe a closed linear subspace of a Hilbert Space H, let xbe a vector not in Mand let dbe the distance from xto M. Then there exists a unique vector y0in Msuch that xy0=d

We define z0=xy0,xH,y0M

y0H
xy0H
z0H
d=dxM=infxm:mM
dxmmMandE17
xm0mM
infxm:mM0
d0

Claim:- d>0,

If d=0then infxm:mM=0.

Then there exists a sequence mnin Msuch that xmn=0as n

mnxas n.

xM, since Mis closed.

This is a contradiction to xM. d0.

Hence d>0.

z0=xy0=d
z0=d>0
z0>0
z00

xMxHsince MH

y0My0H
xy0H
z0H

This proves the existence of non-zero vector z0in H.

We conclude the proof by showing that if yis an arbitrary vector in Mthen z0M.

Let αbe any scalar then

z0αy=xy0αy
=xy0+αy
αz0
z0αyz0
z0αyz0αyz0z0
z0z0z0αyαyz0+αyαyz0z0
α¯z0yαyz0+αα¯yy0E18

Put α=βz0yfor an arbitrary real βthen α¯=βz0,y¯i.e. (18) becomes

βz0,y¯z0yβz0yyz0+βz0yβz0,y¯y2=0
βz0y2βz0y2+β2z0y2y20
2βz0y2+β2z0y2y20
βz0y22+βy20
βz0y2βy220E19

Clearly

z0y=0

Suppose

z0y>0

Choose βarbitrary smallest +ve such that βy2<2

βy22<0
βz0y2βy22<0

This is a contradiction to the Eq. (19)

z0y=0
z0y=0
Thereforez0y,yM
z0M.

Hence it is proved.

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5. Definition 4

5.1 Adjoint of an operator

Let Hbe a Hilbert Space and Tbe an operator on Hthen the mapping T:HHdefined by Txy=xTyx,yHis called the adjoint of T. We verify that Tis actually an operator on H[13]

  1. To prove that Tis linear

    i.e., To prove

    Ty+z=Ty+Tz

    Tαy=αTy

    xTy+z=Txy+z

    =xTy+xTz

    =xTy+Tz

    xTy+z=xTy+Tz

    Ty+z=Ty+Tz

    xTαy=Txαy

    =α¯Txy

    =α¯xTy

    xTy+z=xαTy

    Tαy=αTy

    Therefore Tis linear

  2. To prove that Tis continuous

    0Tx2

    =TxTx

    =TTxx

    =TTxx

    TTxx

    TTxx(by Schwartz Inequality)

    TxTx

    supTx/x1Tif x1

    TT

Since Tis bounded, Tis also bounded.

Hence Tis an operator on H.

These facts tell us that TTis an mapping of BHinto itself

This mapping is called the adjoint operation on BH. [B(H) is a Banach Space].

5.2 Theorem 4.1

The adjoint operation TTon BHhas the following properties [9]

  1. T1+T2=T1+T2

  2. αT=α¯T

  3. T1T2=T2T1

  4. T=T

  5. T=T

  6. TT=T2

5.2.1 Proof

  1. T1+T2=T1+T2

    xT1+T2y=T1+T2xy

    =T1x+T2xy

    =T1xy+T2xy

    =xT1y+xT2y

    xT1+T2y=xT1y+xT2y

    T1+T2y=T1y+T2y

    T1+T2y=T1+T2yyH

    T1+T2=T1+T2

  2. To prove αT=α¯T

    xαTy=αTxy

    =αTxy

    =αTxy

    =αxTy

    xαTy=αxTyxH

    αT=α¯T

  3. To prove T1T2=T2T1

    xT1T2y=T1T2xy

    =T1T2xy

    =T2xT1y

    xT1T2y=xT2T1y

    T1T2y=T2T1yyH

    T1T2=T2T1

  4. To prove that T=T

    xTy=TxyxH

    =y,Tx¯

    =Tyx¯xH

    =xTy

    Ty=Ty

    T=T

  5. To prove that T=T

    0Tx2

    =TxTx

    =TTxx

    =TTxx

    TTxx

    TTxx[by Schwartz Inequality].

    TxTxif x=1].

    supTx/x=1T

    TxT(20)

    We know that T=T.

    T=T

    T

    TT(21)

    From Eq. (20) and (21) we get

    T=T

  6. To prove that TT=T2.

    0Tx2

    =TxTx

    =TxTx

    =TTxx

    =TTxx

    TTxxby Schwartz Inequality]

    TTxx

    Tx2TTx2if x=1

    Tx2TTif x=1

    supTx/x=1TT

    T2TT(22)

    TT=TT

    TT

    T2(23)

    From Eq. (22) and (23) we get.

    T2=TT

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6. Definition 5

6.1 Self Adjoint operator

An operator Aon a Hilbert Space His said to be self Adjoint if A=A. Since 0=0and I=I, 0and Iare self-Adjoint operator [14].

6.2 Theorem 5.1

The Self-Adjoint operator in BHfrom the closed real linear subspace of BHand a real banch space which contains the identity transformation [14].

6.2.1 Proof

We will notice here about the product of two self-adjoint operators.

Let Sdenote the set of all Self-Adjoint operator in BH.

To prove S¯is a real linear subspace of BH¯.

Let

A1,A2S

A1andA2are Self Adjoint.

A1=A1and A2=A2.

Let α,βare real,

αA1+βA2=αA1+βA2
=α¯A1+β¯A2
=αA1+βA2
=αA1+βA2
αA1+βA2=αA1+βA2
αA1+βA2S

Therefore Sis real linear subspace of BH.

Further if Anis a sequence of self Adjoint operators which converges to an operator A. Then it is easy to see that Ais also self Adjoint.

i.e.) Let Anbe a sequence in Ssuch that AnA.

AA=AAn+AnAn+AnA
AAn+AnAn+AnA

AAn+AnAby T=T.

2AnA0as AnA

Therefore AnA0

AnA=0(since norm cannot be –ve)

AA=0
A=A

Ais Self Adjoint

Therefore AS.

Hence Sis closed in BH.

Therefore BHis complete, Sis complete.

Hence Sis a real Banach Space.

I=I,Iis self-Adjoint.

IS[HContains identity transformation].

Hence it is proved.

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7. Conclusion

Even though every Hilbert Space is a Banach space, but there exist plenty of Banach space which are not Hilbert Spaces. However the converse is not true [13]. The Parallelogram Identity gives a criterion for normed space to become an inner product space [15]. It is important to emphasize that every finite dimensional normed Linear Space is a Hilbert Space [2]. Since every finite dimensional normed space is complete. The Theorems of this section allows us to define the adjoint Operator of a bounded Operator [3]. Finally we studied the Self adjoint Operator and its Properties. The results included here are Classical and can be found in the following Reference books in Functional Analysis.

References

  1. 1. Simmons, GF. Introduction to Topology and Modern Analysis. New York: McGraw-Hill, 1963.
  2. 2. Murphy G. C*-Algebras and Operator Theory. San Diego: Academic Press; 1990
  3. 3. Young N. An Introduction to Hilbert Space. Cambridge, England: Cambridge University Press; 1988
  4. 4. Zhu K. An Introduction to Operator Algebras. Boca Raton, USA: CRC Press; 1993
  5. 5. Evan Jenkins B. Representation Theory, Symmetry, and Quantum Mechanics. Chicago, USA: University of Chicago; 2008
  6. 6. Heinrich Saller L. The Basic Physical Lie Operations. Munich, Germany: Max-Planck Institute for Physics; 2004
  7. 7. John DePree S. Introduction to Real Analysis. New York: John Wiley and Sons; 1988
  8. 8. Joseph Maciejko K. Representations of Lorentz and Poincare Groups. Stanford, California: MITP and Department of Physics, Stanford University; 2012
  9. 9. Kreyszig E, Birkho G. The establishment of functional analysis. Historia Mathematica. 1984;11:258-321
  10. 10. Joachim Weidmann M. Linear Operators in Hilbert Space. New York, USA: Springer Karstedt; 1980
  11. 11. Conway J. A Course in Functional Analysis. New York: Springer-Verlag; 1990
  12. 12. James R, editor. Topology (Second Edition). Mungres Eastern Economy Edition. New York, USA: Pearson; 2018
  13. 13. Kreysig E. Introductory Functional Analysis with Applications. New York, USA: John Wiley and Sons; 1978
  14. 14. Debnath L, Mikusinskin P. Introduction to Hilbert Space with Applications. New York, USA: John Wiley and Sons; 2005
  15. 15. Rudvin W. Functional Analysis. New York: MacGraw-Hill; 1973

Written By

Karthic Mohan and Jananeeswari Narayanamoorthy

Submitted: March 20th, 2020 Reviewed: May 8th, 2020 Published: April 7th, 2021