The essential spectrum
The essential spectrum
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基本频谱
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
N. Feldman
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文献类型：
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作者：
A. Aleman;W. Ross;N. Feldman
设B（H）是Hilbert空间（H）上的有界线性算子代数，K是（H）上的紧算子理想，则形成Calkin代数B（H）/K和自然映射π：B（H）→ B（H）/K.回想一下，如果π（A）在B（H）/K中可逆，则A∈B（H）是Fredholm。一个著名的定理[19，p.356]说，A是Fredholm精确地当Rng A是封闭的，并且ker A和H/RangA都是有限维的。算子A是半Fredholm的，如果π（A）在B（H）/K中右可逆或左可逆. Equivalent. A是semi-Fredholm当且仅当RngA是闭的且ker（A）或H/RngA是有限维的。我们还使用符号
$$ sigma（A）：= { lambda in mathbb{C}：lambda I-A is not invertible}（A的谱），$$
$$ sigma（A）：= { lambda in mathbb{C}：lambda I-A is not Fredholm}（A的本质谱）。$$
注意，σ e（A）<$σ（A）。对于半Fredholm算子A，令
$$ ind（A）：= dim ker A - dim（H/Rng A）$$
是A的索引。当集合{±∞|被赋予离散拓扑，映射A→ ind（A）（从半Fredholm算子集到半Fredholm {±∞|是连续的[19，p.361]。
If B(H) is the algebra of bounded linear operators on a Hilbert space (H) and K is the ideal of compact operators on (H), one forms the Calkin algebra B(H)/K and the natural map π: B(H) → B(H)/K. Recall that A∈B(H) is Fredholm if π(A) is invertible in B(H)/K. A well-known theorem [19, p. 356] says that A is Fredholm precisely when Rng A is closed and both ker A and H/RangA are finite dimensional. An operator A is semi-Fredholm if π(A) is either right or left invertible in B(H)/K. Equivalently. A is semi-Fredholm if and only if RngA is closed and either ker(A) or H/RngA is finite dimensional. We also use the notation
$$ sigma (A): = { lambda in mathbb{C}:lambda I - A is not invertible} (spectrum of A),$$
$$ sigma (A): = { lambda in mathbb{C}:lambda I - A is not Fredholm} (essential spectrum of A).$$
Note that σ e (A) ⊂ σ(A). For a semi-Fredholm operator A let
$$ ind(A): = dim ker A - dim (H/Rng A)$$
be the index of A. When the set ℤ∪{±∞| is endowed with the discrete topology, the map A→ ind(A) (from the set of semi-Fredholm operators to ℤ∪{±∞| is continuous [19, p. 361].