Vol 2, No 12 (2011)

(1)

International Journal of Mathematical Archive-2(12), 2011, Page: 2578-2584

Available online through

_{www.ijma.info}

ON

b

-

FRAMES IN BANACH SPACES

M. I. Ismailov*

Baku State University and scientific worker of Institute of Mathematics and Mechanics of NAS Azerbaijan

E-mail: [email protected]

(Received on: 05-12-11; Accepted on: 21-12-11)

___________________________________________________________________________________________________ ABSTRACT

I

n the paper the generalization of the notion

b

-frame in Banach spaces is given by means of bilinear mappings and the notion of

b

-convergence and

b

-representation systems are introduced. The relations between these notions are established. The criterion when the system in Banach space forms

b

-frame, is given. The projection description of

b

-frame is constructed. Using a linear algorithm of expansion in

b

-frame, the extremality estimation for the coefficients of expansion in

b

-frame are established.

Key words:

b

-frame, expansion in

b

-frame,

b

-basis,

b

-completeness ,

CB

-space.

2000 Mathematics Subject Classification: 41A17, 41A58, 42C15.

___________________________________________________________________________________________________

1.INTRODUCTION:

Note that, the notion of frame was determined by Duffin and Schaeffer [1] in Hilbert spaces in the following way. Let

H

be a separable Hilbert space the system of non-zero elements

{ }

ϕ

n _n∈_N

⊂

H

be called a frame in

H

if there exist the

constants

0 <

A

≤

B

<

∞

such that for each

h

∈

H

it is valid

A

h

2_H

≤

∞

=1

2

)

,

(

n

h

ϕ

≤

B

h

2_H , (1)

where

⋅

_H and

(

⋅

,

⋅

)

is a norm and scalar product in

H

. The constants

A

and

B

in (1) are called lower and upper bounds the number

k

=

A

B

is called a condition coefficient of the frame

{ }

ϕ

n _n∈_N. In the case, when

k

=

1 { }

ϕ

n n∈N

is an tight frame.

Development in theory of a frame in Hilbert spaces reduced to obtaining the analogues of the known results for Banach case. By theory of frame [2]-[5] and others have Banach extensions. The notion of

X

-frame generalizing the notion of

p

-frame studied in [5] was introduced in the paper [6]. In [7] gives the dual notion of a -frame in Banach space, for which in [8] projection description is studied, a criterion on the existence of a linear expansion algorithm in frame is proved, the analogue of the extremality properties for the coefficients of expansion in frame is obtained. A frame is a special case of Bessel sequence introduced and studied [9] in Hilbert spaces. Banach analogies these results were investigated in the paper [10]-[14].

Cite the known result [15] from the quant theory of information for projection description of frame.

Theorem: 1.1 Let

{ }

ϕ

_n _n_∈_N be a tight frame in Hilbert space

H

. Then there exists a Hilbert space

H

′ ⊃

H

and orthonormed basis

{ }

ψ

_n _n_∈_N

⊂

H

′

such that

)

(

ψ

_n

π

=

ϕ

n,

n

∈

N

,

where

π

is an operator of orthogonal projection from

H

′

onto

H

.

________________________________________________________________________________________________

(2)

b

!" # $!% &'( &()

© 2011, IJMA. All Rights Reserved 2579 The results [16], [17] and [18] in this direction are known. The frames in Hilbert spaces have close relations with Riesz bases. This matter was considered [19] and is in the following.

Theorem: 1.2 For the system

{ }

ϕ

_n _n_∈_N of Hilbert space

H

to be a frame with boundaries

A

and

B

,

A

≤

B

, it is necessary and sufficient to find a Hilbert space

H

′

containing

H

and the Riesz basis

{ }

ψ

_n _n_∈_N

⊂

H

′

such that

2

1

A

2 1

1 2 ∞

=

n n

c

≤

H n

n n

c

′ ∞

=1

ψ

≤

_B

12

2 1

1 2 ∞

=

n n

c

, (2)

where

π

(

ψ

_n

)

=

ϕ

n,

n

∈

N

,

π

is an operator of orthogonal projection from

H

′

onto

H

.

The goal of our paper is extension of the results of [8] to

b

_X~-frames for

CB

-space

X

~

. In the paper, necessary and

sufficient condition

b

_X~-frame of the system in Banach space is proved, projection description of

X

b

~-frame and criterion of

the existence of linear expansion algorithm in

b

_X~-frame is given.

2.

b

-FRAME IN BANACH SPACE:

Let

X

be a Banach space with the norm

⋅

_X . Denote by

X

~

B

-space of sequences

~

x

=

{ }

x

n _n∈_N,

x

n

∈

X

, with

coordinate-wise linear operations. If the convergence in

X

~

is coordinate-wise and the subspaces

n

E

~

=

{

x

∈

X

:

~

x

=

{

in

x

}

_i∈_N

,

x

∈

X

}

~

_δ

_{form a basis, then}

_X

~

_called

_CB

_{-space with a canonical basis.}

Find the general form of a linear continuous functional in

X

~

. Let

~

t

be an arbitrary linear continuous functional in

X

~

.

Since for each

x

~

∈

X

~

(

x

~

=

{ }

x

n _n∈_N) the equality

x

~

=

{

}

∞

=

∈ 1

n

N i n in

x

δ

is valid, then

~

t

(

~

x

)

=

{

}

∞

=

∈ 1

)

(

~

n

N i n in

x

t

δ

.

Consider the functional

t

_n

:

X

→

C

such that

t

_n

(

x

)

=

~

t

(

{

δ

_in

x

}

_i_∈_N

)

for

∀

x

∈

X

. Then we get that

)

~

(

~

x

t

=

∞

=1

)

(

n n n

x

t

. From the coordinate-wise convergence in

X

~

it follows that

t

n

∈

X

*. Consequently, identifying

t

~

with

{ }

t

_n _n_∈_N

⊂

*

X

,

X

~

* is isometrically isomorphic to some

B

-space of sequences of elements from

X

* and

~

t

(

~

x

)

=

∞

=1

)

(

n n n

x

t

is a general form of a linear continuous functional in

X

~

, with the norm

*

~

X

t

=

1 ~ _~

sup

≤

X x

∞

=1

)

(

n n n

x

t

.

Let

Y

and

Z

be

B

-spaces with appropriate norms _Y and _Z. Consider a bilinear mapping

b

(

x

,

y

)

:

X

×

Y

→

Z

satisfying the condition:

∃

M

0

:

Z

xy

≤

M

x

_X

y

_Y

∀

x

∈

X

,

y

∈

Y

. (3)

where

xy

=

b

(

x

,

y

)

,

x

∈

X

,

y

∈

Y

. Denote by

b

ˆ

(

f

,

y

)

the mapping

b

ˆ

(

f

,

y

)

:

Z

*

×

Y

→

X

*, determined by the relation

b

ˆ

(

f

,

y

)(

x

)

=

f

(

xy

)

∀

f

∈

Z

*,

y

∈

Y

,

x

∈

X

. Further, for brevity we assume

<

f

,

y

>

=

b

ˆ

(

f

,

y

)

. It is easy to get from (3) the validity of the inequality

<

f

,

y

>

_X*

≤

M

f

_Z*

y

_Y

*

Z

f

∈

(3)

b

!" # $!% &'( &()

X

~

be a

CB

-space with a canonical basis. Introduce the following denotation.

Definition: 2.1 The system

{ }

ϕ

_n _n_∈_N

⊂

Y

is said to be

X

b

~-frame in

Z

if there exist the constants

A

,

B

,

∞

<

≤

<

A

B

0

such that for any

g

∈

Z

* the sequence

{

<

g

,

ϕ

_n

>

}

_n_∈_N belongs to

X

~

* and satisfies the inequality

A

g

_Z*

≤

{

<

g

,

ϕ

n

>

}

n∈N _X~*

≤

B

g

_Z*. (5)

Definition: 2.2 The system

{ }

ϕ

_n _n_∈_N

⊂

Y

is said to be a system of

b

_X~-representation if for any

z

∈

Z

there exists

{ }

x

n _n∈_N

∈

Z

such that

z

=

∞

=1

n n n

x

ϕ

and a system of

X

b

~-convergence in

Z

if for any

~

x

∈

X

~

(

x

~

=

{ }

x

_n _n_∈_N) the

series

∞

=1

n n n

x

ϕ

converges in

Z

.

Cite the criteria when the system

{ }

ϕ

_n _n_∈_N

⊂

Y

forms a

b

_X~-frame in

Z

.

Theorem: 2.3 The system

{ }

ϕ

_n _n_∈_N

⊂

Y

forms a

b

_X~-frame in

Z

iff it is simultaneously both a system of

b

_X~

-representation and a system of

X

b

~-convergence in

Z

.

Proof: Necessary: Let

{ }

ϕ

_n _n_∈_N be a

b

_X~-frame in

Z

. Determine the operator

S

(of synthesis) in linearity on finite

sequences

{ }

x

_n by the equality

S

(

{ }

x

_n

)

=

n n n

x

ϕ

. For an arbitrary

g

∈

Z

*we get

{ }

))

(

S

x

_n

g

=

n n n

x

g

(

ϕ

)

=

(

)

n

n n

x

g

ϕ

=

,

(

_n

)

n

x

g

>

<

ϕ

=

{

<

g

,

ϕ

n

>

}

_n∈_N

(

{ }

x

n

)

.

Then

{ }

))

(

S

x

_n

g

=

|

{

<

g

,

ϕ

n

>

}

_n∈_N

(

{ }

x

n

)

|

≤

{

<

g

,

ϕ

n

>

}

n∈N _X~*

{ }

x

n X~

≤

B

g

Z*

{ }

x

n X~.

So,

S

(

{ }

x

n

)

_Z

≤

B

{ }

x

n _X~. Continuing

S

in continuity on all

X

~

, we get

S

∈

L

(

X

~

,

Z

)

and

S

(

~

x

)

=

∞

=1

n n n

x

ϕ

,

X

x

~

_∈

₍

_x

~

₌

_{{ }}

N n n

x

_∈ ). Consequently,

{ }

ϕ

_n _n_∈_N is a system of

b

_X~-convergence in

Z

. Consider an operator (of

analysis)

R

:

Z

*

→

~

*

X

in the equality

R

(

g

)

=

{

<

g

,

ϕ

n

>

}

_n∈_N, *

Z

g

∈

. It is clear that it holds the equality

R

=

S

*. Since for any

g

∈

Z

* it holds

A

g

_Z*

≤

~_*

*

)

(

X

g

S

, then

S

maps

X

~

on all the space

Z

(see [20], theorem 4.15).

Thus,

{ }

ϕ

_n _n_∈_Nis a system of

b

_X~-representation in

Z

.

Sufficiency: Let

{ }

ϕ

_n _n_∈_N be a system of

b

_X~-representation and

b

_X~-convergence in

Z

. Consider (linear) operators

n

S

:

X

~

→

Z

by the equalities

S

_n

(

~

x

)

=

n

k k k

x

1

ϕ

,

n

∈

N

. From the coordinate-wise convergence in

X

~

it follows,

that

S

_n

∈

L

(

X

~

,

Z

)

. Since

{ }

ϕ

_n _n_∈_N is a system of

b

_X~-convergence in

Z

, then

∃

∞ →

n

lim

S

_n

(

~

x

)

and by the theorem on

uniform boundedness

) , ~ (X Z L n

S

≤

B

for any

n

∈

N

. Consequently, the operator

S

determined by the equality

)

~

(

x

S

=

∞ →

n

lim

S

_n

(

~

x

)

belongs to

L

(

X

~

,

Z

)

. Further, consider an operator

R

∈

L

(

Z

*

,

X

~

*

)

such that

R

=

*

S

. Then for

any

g

∈

Z

* and

~

x

∈

X

~

we get

)

(

g

R

(

~

x

)

=

S

*

(

g

)

(

~

x

)

=

g

(

S

(

~

x

))

=

∞

=1

)

(

n

n n

x

(4)

b

!" # $!% &'( &()

R

(

g

)

=

{

<

g

,

ϕ

n

>

}

_n∈_N. Since

{ }

ϕ

n n∈N is a system of

b

X~-reresentation then

S

X

)

=

Z

~

(

. Therefore

there exists

A

>

0

:

A

g

_Z*

≤

~*

*

)

(

X

g

S

(see. [20], theorem 4.15) and

{ }

ϕ

_n _n_∈_N forms a

b

_X~-frame in

Z

. The

theorem is proved.

3. PROJECTIVE

b

-FRAMES:

Let

X

, Y and

Z

be

B

-spaces , a

X

~

be a

CB

-space with a canonical basis. Cite some notion from [14].

The system

{ }

ϕ

_n _n_∈_N

⊂

Y

is said to be

b

-complete in

Z

if the aggregate

L

_b

(

{ }

ϕ

_n _n_∈_N

)

of all possible finite sums

n n

x

ϕ

is dense in

Z

.

The system

{ }

ϕ

_n _n_∈_N

⊂

Y

is said to be

b

_X~-basis in

Z

if for

∀

z

∈

Z

there exists a unique sequence

{ }

x

_n _n_∈_N

⊂

X

:

z

=

∞

=1

n n n

x

ϕ

, moreover

X

~

=

{

~

x

=

{ }

x

n _n∈_N

⊂

X

: ∞

=1

n n n

y

x

∈

Z

}

.

As the analogue of the projective frame (see [8]) introduce the following definition.

Definition: 3.1

b

-frame

{ }

ϕ

_n _n_∈_N

⊂

Y

in

Z

is said to be projective if: (à) there exist

B

-space

Z

~

⊃

Z

, including the space

Z

as a closed subspace; (b) there exist

b

_X~

~

-basis

{ }

ψ

_n _n_∈_N

⊂

L

(

X

,

Z

~

)

in

Z

~

, where

b

~

(

x

,

ψ

)

≡

(

x

,

ψ

)

=

ψ

(

x

)

,

x

∈

X

,

ψ

∈

L

(

X

,

Z

~

)

; (c) there exist a continuous projector

P

∈

L

(

Z

~

,

Z

)

from

Z

~

onto

Z

such that

P

(

x

,

ψ

_n

)

=

x

ϕ

n for

∀

x

∈

X

,

N

n

∈

.

Definition: 3.2 The space of coefficients of zero-series of the system

{ }

ϕ

_n _n_∈_N

⊂

Y

is called a closed in

X

~

subspace

N

~

=

{

x

~

=

{ }

x

n _n∈_N

∈

X

~

:

∞

=1

n n n

x

ϕ

=

0 }

.

Theorem: 3.3 Let

{ }

ϕ

_n _n_∈_N

⊂

Y

be a

X

b

~-frame in

Z

. Then

{ }

ϕ

_n _n_∈_N is a projective

X

b

~-frame in

Z

iff

N

~

is

complemented in

X

~

.

Proof: Necessity: Let

{ }

ϕ

_n _n_∈_N be a projective

b

_X~-frame in

Z

. By theorem 2.3 there exists a synthesis

operator

S

∈

L

(

X

~

,

Z

)

:

S

(

~

x

)

=

∞

=1

n n n

x

ϕ

,

x

~

∈

X

~

(

~

x

=

{ }

x

_n _n_∈_N). Let

J

be a natural isomorphism between

X

~

and

Z

~

, i.e.

J

(

{ }

x

_n _n_∈_N

)

=

∞

=1

)

,

(

n

n n

x

ψ

and

P

be a projector from

Z

~

onto

Z

. Then the following equality holds:

PJ

S

=

. Denote by

M

~

=

J

−1

(

Z

)

a closed subset in

X

~

. Show that

X

~

=

M

~

⊕

N

~

. Take an arbitrary

~

x

from

X

~

. Denote by

z

=

S

(

~

x

)

and

~

x

_M~

=

J

−1

(

z

)

. Consider

x

~

_N~

=

x

~

−

~

x

_M~. Since

S

(

~

x

_N~

)

=

S

(

~

x

)

−

S

(

~

x

_M~

)

=

0

, then

N

x

_N

~

∈

, consequently,

~

x

=

M

x

~

₊

N

x

~

_{. Now let}

~

_x

_∈

_M

~

_∩

_N

~

_{. From}

~

_x

_∈

_N

~

_{it follows that}

_S

₍

~

_x

₎

₌

₀

_{. Since}

x

~

_∈

_M

~

_{, there exists}

_z

_from

_Z

_{, such that} 1

₍

₎

z

J

−

=

~

x

. Consequently,

z

=

P

(

z

)

=

PJ

(

~

x

)

=

S

(

~

x

)

=

0

Therefore

x

~

₌

1

₍

₎

z

J

−

=

0

. Thus,

X

~

=

M

~

⊕

N

~

.

Sufficiency: Let

N

~

be complemented in

X

~

, i.e.

X

~

=

M

~

⊕

N

~

. Denote by

Z

~

the direct sum of

Z

⊕

N

~

. Obviously,

Z

~

is a

B

-space by the norm

{ }

z

,

x

~

_Z~

=

max

(

)

X Z

x

(5)

b

!" # $!% &'( &()

Z

, i.e.

P

(

{ }

z

,

~

x

)

=

z

, and

Q

a projection operator from

X

~

onto

M

~

along

N

~

. Define the operator

J

:

X

~

→

Z

~

from the formula

J

(

~

x

)

=

{

S

(

~

x

),

~

x

−

Q

(

~

x

)

}

. Obviously,

J

∈

L

(

X

~

,

Z

~

)

. Show that

J

is an isomorphism between

X

~

and

Z

~

. If

J

(

~

x

)

=

0, so

S

(

~

x

)

=

0 and

~

x

=

Q

(

~

x

)

. Then

~

x

∈

M

~

∩

N

~

, therefore

x

~

=

0

. Further, we

have

J

(

X

~

)

=

Z

~

. Indeed, if

{

z

,

x

~

₀

}

∈

Z

~

, then there exists

ξ

~

∈

X

~

such that

z

=

S

(

ξ

~

)

. Denote

~

x

=

~

x

₀

+

Q

(

ξ

~

)

. It is clear that

Q

(

~

x

)

=

Q

(

ξ

~

)

. So,

~

x

₀

=

~

x

−

Q

(

~

x

)

and

z

=

S

(

ξ

~

)

=

S

(

Q

(

ξ

~

))

=

S

(

Q

(

x

~

))

=

S

(

~

x

)

,

i.e.

{

z

,

~

x

₀

}

=

{

S

(

~

x

),

~

x

−

Q

(

x

~

)

}

. Using the Banach theorem on inverse operator, we get that

J

is an isomorphism. Since

X

~

is a

CB

- space with a canonical basis, the system

{ }

ψ

_n _n_∈_N:

J

(

{

δ

_in

x

}

_i_∈_N

)

=

(

x

,

ψ

_n

)

forms

b

_X~

~

-basis in

Z

~

.

On the other hand for

∀

x

∈

X

,

n

∈

N

it is valid

P

(

x

,

ψ

n

)

=

x

ϕ

n.

The theorem is proved.

4.LINEAR EXPANSION ALGORITHM IN X

b

~-FRAME:

Let

X

,

Y

and

Z

be

B

-spaces,

X

~

be a

CB

-space with a canonical basis.

Definition: 4.1We say that it holds a linear expansion algorithm in

b

_X~-frame

{ }

ϕ

_n _n_∈_N

⊂

Y

in

Z

, if there exists a system

{ }

n n∈N

*

ϕ

⊂

L

(

Z

,

X

)

such that for any

z

∈

Z

the sequence

{

ϕ

_n*

(

z

)

}

_n_∈_N belongs to

X

~

and

z

=

∞

=1 *

)

(

n

n n

z

ϕ

.

Theorem: 4.2Let

{ }

ϕ

_n _n_∈_N

⊂

Y

be

X

b

~-frame in

Z

. For the existence of a linear expansion algorithm in

X

b

~-frame

{ }

ϕ

n n∈N it is necessary and sufficient that

{ }

ϕ

n n∈N be a projective

b

X~-frame.

Proof:Necessity: Let there exists a linear expansion algorithm in

b

_X~-frame

{ }

ϕ

_n _n_∈_N. As in the proof of theorem 2.1 we

denote by

S

∈

L

(

X

~

,

Z

)

:

S

(

~

x

)

=

∞

=1

n n n

x

ϕ

,

~

x

∈

X

~

(

~

x

=

{ }

x

n _n∈_N) the synthesis operator in

X

~

. Determine the

operator

T

:

Z

→

X

~

from the formula

T

(

z

)

=

{

n

(

z

)

}

_n∈_N *

ϕ

. Show that

T

∈

L

(

Z

,

X

~

)

. Consider the sequence of the

operators

T

_m

∈

L

(

Z

,

X

~

)

:

T

_m

z

=

{

}

=

∈

m

n

N i n in

z

1 *

)

(

ϕ

δ

. Since the series

{

}

∞

=

∈ 1

*

)

(

n

N i n in

ϕ

z

δ

converges, there exists

∞ →

m

lim

T

m

z

and by the Banach Steinhous theorem the sequence

{ }

T

n n∈N is bounded. So,

T

∈

)

~

,

(

Z

X

L

, since

Tz

=

∞ →

m

lim

T

m

z

. Obviously

(

ST

)(

z

)

=

(

)

* 1

z

n n

ϕ

∞

=

n

ϕ

=

z

, i.e.

ST

=

I

, where

I

is an identity operator in

Z

.

Further, let

M

~

=

T

(

Z

)

. We have

X

~

=

M

~

⊕

N

~

, where

N

~

=

KerS

. Indeed, let

x

~

be an arbitrary element in

X

~

. Denote

z

=

S

(

~

x

)

and

x

~

_M~

=

T

(

z

)

. Then

S

(

~

x

−

~

x

_M~

)

=

S

(

~

x

)

−

S

(

~

x

_M~

)

=

0

. Therefore

~

x

_N~

=

x

~

−

x

~

_M~

∈

N

~

.

Show that

M

~

∩

N

~

=

{ }

0

. If

~

x

∈

M

~

∩

N

~

, then

S

(

~

x

)

=

0

. Suppose that

z

from

Z

is such that

T

(

z

)

=

x

~

. We get

z

=

(

ST

)(

z

)

=

S

(

~

x

)

=

0

. Consequently,

x

~

=

0

and

X

~

=

M

~

⊕

N

~

. It remains to apply theorem 3.3.

Sufficiency: Let

{ }

ϕ

_n _n_∈_N be a projective

b

_X~-frame in

Z

. Then by theorem 3.3

N

~

is complemented in

X

~

, i.e.

(6)

b

!" # $!% &'( &()

)

(

)

(

z

J

1

z

T

=

− ,

z

∈

Z

, and operator

e

_n(

X

~

→

X

):

e

_n

(

~

x

)

=

x

_n,

~

x

=

{ }

x

_n _n_∈_N. Determine the operator

ϕ

_n* :

Z

→

X

from the expression

ϕ

*n

(

z

)

=

e

n

(

T

(

z

))

. So,

Tz

=

{

n

(

z

)

}

n∈N

*

ϕ

. Take an arbitrary

z

∈

Z

. Let

x

~

=

T

(

z

)

(

x

~

=

{ }

x

n _n∈_N). Then we have

z

=

(

ST

)(

z

)

=

S

(

~

x

)

=

∞

=1

n n n

x

ϕ

=

*

(

)

1

z

n n

ϕ

∞

=

n

ϕ

.

The theorem is proved.

Theorem: 4.3Let

{ }

ϕ

_n _n_∈_N

⊂

Y

be a

X

b

~-frame in

Z

and

N

~

be a space of coefficients of zero-series of the system

{ }

ϕ

n n∈N. Then the following properties are equivalent:

(i) there exists a continuous projector

P

:

X

~

→

N

~

, such that

I

−

P

=

1

;

(ii) it holds a linear expansion algorithm in

{ }

ϕ

_n _n_∈_N, moreover if for

z

∈

Z

, there exists the sequence

{ }

x

_n _n_∈_N from

X

~

:

z

=

∞

=1

n n n

x

ϕ

, then

{ }

X N n n

x

_∈ ~

≥

{

}

X N n n

z

~ *

)

(

_∈

ϕ

.

Proof: Let condition (i) be fulfilled. Then

N

~

is complemented in

X

~

, i.e.

X

~

=

M

~

⊕

N

~

, where

M

~

=

KerP

(see [20], theorem 5.1). Consequently,

{ }

ϕ

_n _n_∈_N is a projective

b

_X~-frame in

Z

and by theorem 4.2 it holds a linear

expansion algorithm. Take an arbitrary

z

∈

Z

. Suppose that there exists

{ }

x

_n _n_∈_N

∈

X

~

, such that

z

=

∞

=1

n n n

x

ϕ

. Let

S

be a synthesis operator,

J

X

~

and

Z

~

,

Q

=

I

−

P

,

T

be a contraction of

1 −

J

on

Z

. Since

X

~

=

M

~

⊕

N

~

, then

x

~

=

x

~

_M~

+

~

x

_N~. Let

u

∈

Z

:

~

x

_M~

=

T

(

u

)

. We have

u

=

(

ST

)(

u

)

=

(

~

)

M

x

S

=

S

(

~

x

)

=

z

.

Consequently,

{

}

X N n n

z

~ *

)

(

_∈

ϕ

=

T

(

z

)

_X~

=

X M

x

~ ~

~

₌

X

x

Q

(

~

)

~

≤

{ }

X N n n

x

_∈ ~.

Let now condition (ii) be fulfilled. Then by theorem 4.2 the

b

_X~-frame

{ }

ϕ

_n _n_∈_N is projective and so by theorem 3.3 the

subspace

N

~

is complemented in

X

~

, i.e.

X

~

=

M

~

⊕

N

~

. As above, let

S

be a synthesis operator,

J

X

~

and

Z

~

=

Z

⊕

X

~

and

T

∈

L

(

Z

,

X

~

)

be an operator determined by the expression

)

(

z

T

=

{

n

(

z

)

}

_n∈_N *

ϕ

.

Denote by

Q

the operator projecting

X

~

on

M

~

along

N

~

and

P

=

I

−

Q

. Take an arbitrary

x

~

∈

X

~

. Obviously, there exists

~

x

_N~

=

~

x

−

Q

(

x

~

)

∈

N

~

. Since

Q

(

~

x

)

∈

M

~

, then there exists

z

∈

Z

such that

T

(

z

)

=

Q

(

~

x

)

, since

M

Z

T

(

)

=

~

. We have

z

=

(

ST

)(

z

)

=

S

(

Q

(

~

x

))

=

S

(

~

x

)

=

∞

=1

n n n

x

ϕ

.

According to the condition we get

{ }

X N n n

x

_∈ ~

≥

{

}

X N n n

z

~ *

)

(

_∈

ϕ

. Consequently,

{ }

x

n _n∈_N _X~

≥

T

(

z

)

X~

=

Q

x

)

X~

~

(

,

(7)

b

!" # $!% &'( &()

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p

-Frames in separable Banach spaces, Adv. Comput. Math., 18:2 (2003), 117-126.

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b

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