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This paper investigates the structure of general affine subspaces of <i>L</i><sup>2</sup>(R<i><sup>d</sup></i>) . For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a purely non-reducing subspace, and every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces when |detA| = 2.

Let A be a d × d expansive matrix. Define the dilation operator D and the shift operator T_{k},

respectively. It is easy to check that they are both unitary operators on

In this case, we say that

M. Therefore an affine subspace X of

Lemma 1. Let X and Y be closed subspaces of a Hilbert space H and

1)

2)

Proof. 1) Obviously,

then

So

2) For

since

Lemma 2. Let

1) If

2) If

Proof. We only prove 1) since 2) can be obtained similarly. Since

If

The proof is completed.

Proposition 1. Suppose that X is an affine subspace of _{1} in X and a reducing subspace Y of

in X such that the length of M_{1} is no more than that of M and

Proof. For each

Obviously,

[

Suppose that for some subset

Since

which shows that M_{1} is a shift invariant subspace of length no more than the length of M. The proof is completed.

Proposition 2. Suppose that X is a non-zero affine subspace of

1)

2)

Proof. 1): Obviously,

2): By 1) and Lemma 2, it follows that

If X is purely non-reducing, then

Proposition 3. Let X be an affine subspace of

Proof. We first show that

Next we will show that

So

Lemma 3. Let X and Y be affine subspaces of

Proof. Since

The proof is completed.

Lemma 4. Assume

Proof. Since

which shows that

Lemma 5. Let X be an affine subspace of

1)

2)

3)

Proof. 1): Note that we only need to show

2): Since Q is shift invariant and

If X is a reducing subspace, then

3): By 1) and 2), we have

Proposition 4. Let X and Y be affine subspaces of

Proof. Let M be a shift invariant subspace contained in

Proposition 5. Let X and Y be affine subspaces of

shift invariant subspaces contained in X and Y respectively. Define

Proof. According to Proposition 4,

Write

Hence

due to the fact that

So

Proposition 6. Let X and Y be two affine subspaces of

1)

2)

Proof. 1): By Lemma 5,

due to the facts that

Observe that

2): According to Proposition 5, it follows that

which shows that

Theorem 7. Let X be an affine subspace of

1) There exist a shift invariant subspace M in X such that

2) If X is a non-zero reducing subspace and_{1} and X_{2} such that

3) If X is non-zero and not reducing, then there exists a unique decomposition _{1} be reducing and X_{2} being purely non-reducing;

4) If X is non-zero and

Proof. 1): By Proposition 1, it follows that_{1} is some shift invariant subspace in X and Y is a reducing subspace. If_{1} in the proof of Proposition 1, it follows that

2): Let

Indeed, for

Then

Then_{1} is a purely non-reducing affine subspace. Write

Obviously Q is a shift invariant subspace contained in X_{1} and_{1}. Also by Proposition 3, it is

enough to show

Then for each

since

Hence

Thus_{1} is a purely non-reducing affine subspace. Similarly to X_{2}.

3): Let X be a non-reducing affine subspace of _{1} be the maximal reducing subspace contained in X. Write_{2} is affine by Proposition 6 and X_{2} is purely non-reducing since X_{1} is the maximal reducing subspace in X. Also note that the orthogonal complement of a reducing space within another reducing space is always reducing. Then the uniqueness follows.

4): 4) follows after 2) and 3). The proof is completed.

We thank the Editor and the referee for their comments. This work is funded by the National Natural Science Foundation of China (Grant No. 11326089), the Education Department Youth Science Foundation of Jiangxi Province (Grant No. GJJ14492) and PhD Research Startup Foundation of East China Institute of Technology (Grant No. DHBK2012205).