Switching default male pronoun to default female, in deference to Wizards’ own style and to match rules quotes.
Her wizard class features give her no special ability to activate the item
As she does not have access to Enchantment or Evocation spells, she does not qualify as having those spells on her spell list for the purposes of activating spell-completion and spell-trigger items (e.g. scrolls and wands), even for spells that are otherwise on the wizard spell list. The School Specialization sidebar makes that much, at least, quite clear:
Spells of the prohibited school or schools are not available to the wizard, and she can’t even cast such spells from scrolls or fire them from wands. (PHB1 p57)
However, Use Magic Device is all about using items you usually can’t
A wizard could use the Use Magic Device skill to activate such items, the same as any other character. Wizards don’t get Use Magic Device in-class and don’t tend to invest heavily in Charisma, but the loremaster prestige class is really easy for a wizard to enter, and would get you Use Magic Device.
Anyway, that a wizard can Use Magic Device on spell-completion and spell-trigger items is, unfortunately, not immediately clear from the rule I quoted: that rule says they are not available, with no qualifications. Unfortunately, Wizards had this tendency, particularly early on, to assume single-classed characters, and ignored cross-class skills. It probably simplified the language, but at the cost of making things confusing when you stepped away from those assumptions.
For this, I am going to draw parallels to the question of whether or not a multiclass specialist wizard may cast spells of her banned school with her other class. The books explicitly allow this (Complete Arcane 185), despite that very same rule I’ve quoted above which seems to say that the specialist wizard may not cast those spells at all. What that line actually means is that the wizard class itself does not give you the ability to cast those spells or use those items when you specialize and ban those schools. You can still use the usual methods around that inability, the same as, say, a monk could.
Liches do not require access to the school of Necromancy; the only requirement to become a lich is to have a phylactery, and that only requires Craft Wondrous Item, the ability to cast spells (any kind of spells), and caster level 11th (and a lot of money and XP).
Liches also do not have special affinity for necromancy; their magical talents apply equally-well to all schools of magic.
In fact, the word “necromancy” does not appear anywhere in the Lich monster entry on the SRD.
But anyway, getting spells from a banned school as a wizard can be done. A wizard cannot use items of banned spells without Use Magic Device, but he can use UMD. Eternal wands from Magic Item Compendium don’t even require that a spell is in your spell list, so those shouldn’t even require UMD. Banning schools as a wizard also doesn’t affect other classes, so multiclassing is an option. But neither of these is casting the spell as a wizard.
For that, options become somewhat more obscure:
Recaster (Races of Eberron) and wyrm wizard (Dragon Magic) are prestige classes that let you learn spells from other spell lists entirely. No restriction based on banned schools is mentioned.
- Rainbow servant (Complete Divine) goes further, letting you learn all cleric spells, again without reference to being limited by school banning.
The Spell Reprieve feat (Lost Empress of Faerûn) lets you learn one spell from a banned school.
- The Arcane Transfiguration feat from the same source requires Spell Reprieve and Item Reprieve, so it's extremely expensive, but it does allow you to open up an entire school you had previously banned.
The Diversified Casting feat (Dragon vol. 325) lets you cast three spells from a banned school.
Best Answer
Yes, you can. Specialization is a Wizard class feature that affects Wizard spellcasting only. It has no effect on spells from other classes. Complete Arcane, for example, explicitly suggests multiclassing with Sorcerer to get your missing spells.
The FAQ (which is not a real rules source but may be good for convincing DMs) explicitly answered this question as well.
Specialization would be much clearer if they simply said that the spells were removed from your wizard spell list, and even effects that add spells to your spell list cannot add spells of your banned schools to your wizard spell list. That's what all the specialization rules sum up to, they just chose a long and complicated way to describe it.
However, note that Wizards, like all spellcasters, multiclass poorly. Being behind on spell levels is a very serious blow to your power. The addition of lower-level spells from another class is not nearly enough to justify this, from a purely-mechanical perspective. There are exceptions, but they involve dual-progression PrCs that can be entered with only one level lost, and that have their own class features (like Anima Mage from Tome of Magic or Ultimate Magus from Complete Mage).